Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying
T2=1. Without extra symmetry, it has been found that
Zp parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors
n of
p. In the presence of TR symmetry, however, the enriched classification is characterized by three indices
n, $\ensuremath{\kappa}$, and $\ensuremath{\mu}$, where $\ensuremath{\kappa}\ensuremath{\in}{\mathbb{Z}}_{2}$ denotes the linear or projective TR actions on the edges, and $\ensuremath{\mu}\ensuremath{\in}{\mathbb{Z}}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the
Zr-symmetric parafermionic ground states, where
r=p for trivial or topological phases, and
r=p/n for SSB phases, each original gapped phase with odd
r is divided into two phases, while each phase with even
r is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.
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.
We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies
T2=1. Such chains can be realized in one-dimensional structures embedded in fractionalized two-dimensional states of matter, e.g., at the edges of a fractional quantum spin Hall system, where counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of
Zm parafermions can belong to one of several distinct phases. We find that when the modified time-reversal symmetry is imposed, the classification becomes richer. If
m is odd, each of the phases splits into two subclasses. We identify the symmetry-protected phase as a Haldane phase that carries a Kramers doublet at each end. When
m is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of
m. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.
Abstract A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z 2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z n symmetry, and prove that for appropriate couplings they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.
In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with
Zp parafermions. Unlike the
Z2 Majorana fermions, the
Zp parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible
Z3 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
Z3 parafermion chain and two-coupled parafermion chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with
Zp parafermions within the matrix product representation in one dimension.
Recently, it has been proposed that exotic one-dimensional phases can be realized by gapping out the edge states of a fractional topological insulator. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. We introduce a classification scheme for the phases that can occur in parafermionic chains. We find that the parafermions support both topological symmetry fractionalized phases as well as phases in which the parafermions condense. In the presence of additional symmetries, the phases form a non-Abelian group. As a concrete example of the classification, we consider the effective edge model for a $\ensuremath{\nu}=1/3$ fractional topological insulator for which we calculate the entanglement spectra numerically and show that all possible predicted phases can be realized.