Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1701.01133 · Physical review. B./Physical review. B

Classification of topological phases of parafermionic chains with symmetries

Dganit Meidan, Erez Berg, Ady Stern

Published May 4, 2017• 10 citations
arXivHTMLPDF

Abstract

We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies ${\mathcal{T}}^{2}=1$. 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 ${\mathbb{Z}}_{m}$ 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.

DOI
10.1103/physrevb.95.205104

Related papers

Enriched classification of parafermionic gapped phases with time-reversal symmetry

Wen-Tao Xu, Guang-Ming Zhang

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 ${T}^{2}=1$. Without extra symmetry, it has been found that ${\mathbb{Z}}_{p}$ 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 ${\mathbb{Z}}_{r}$-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.

arXiv:1709.04657•Physical review. B./Physical review. B•Mar 21, 2018•2 citations

Topological phases in gapped edges of fractionalized systems

Johannes Motruk, Erez Berg, Ari M. Turner +1 more

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.

arXiv:1303.2194•Physical Review B•Aug 19, 2013•60 citations

Fermionized parafermions and symmetry-enriched Majorana modes

Aaron Chew, David F. Mross, Jason Alicea

Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate ${\mathbb{Z}}_{4}$ parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.

arXiv:1802.04809•Physical review. B./Physical review. B•Aug 27, 2018•41 citations

Topological Phases of Parafermions: A Model with Exactly Solvable Ground States

Fernando Iemini, Christophe Mora, Leonardo Mazza

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.

arXiv:1611.00832•Physical Review Letters•Apr 25, 2017•39 citations

Parafermionic edge zero modes in Zn-invariant spin chains

Paul Fendley

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.

arXiv:1209.0472•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2012•373 citations

Majorana modes meet fractional fermions in one dimension

Dan-Bo Zhang, Qiang-Hua Wang, Z. D. Wang

Majorana modes and fractional fermions are two types of edge zero modes appearing separately in topological superconductors and dimerized chains. Here we reveal how to harvest both types of edge modes simultaneously in an exotic chain. Such modes are naturally spin-charge separated, and are protected by the inversion and spin-parity symmetries. We construct a lattice model to illustrate the nature of these edge modes, utilizing fermionic functional renormalization group, mean-field theory and bosonization. We also elucidate that the four-fold degenerate ground states with edge-spinons in the Haldane phase of spin-$1$ chain may be reinterpreted as our spin-charge separated edge modes in an equivalent spin-$1/2$ fermionic model.

arXiv:1602.01267•Feb 3, 2016•0 citations
We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies T2=1{\mathcal{T}}^{2}=1T2=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{\mathbb{Z}}_{m}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 mmm 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 mmm is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of mmm. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.
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{T}^{2}=1T2=1. Without extra symmetry, it has been found that Zp{\mathbb{Z}}_{p}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 nnn of ppp. In the presence of TR symmetry, however, the enriched classification is characterized by three indices nnn, $\ensuremath{\kappa}$, 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{\mathbb{Z}}_{r}Zr​-symmetric parafermionic ground states, where r=pr=pr=p for trivial or topological phases, and r=p/nr=p/nr=p/n for SSB phases, each original gapped phase with odd rrr is divided into two phases, while each phase with even rrr 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.
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.
Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z4{\mathbb{Z}}_{4}Z4​ parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
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.
Majorana modes and fractional fermions are two types of edge zero modes appearing separately in topological superconductors and dimerized chains. Here we reveal how to harvest both types of edge modes simultaneously in an exotic chain. Such modes are naturally spin-charge separated, and are protected by the inversion and spin-parity symmetries. We construct a lattice model to illustrate the nature of these edge modes, utilizing fermionic functional renormalization group, mean-field theory and bosonization. We also elucidate that the four-fold degenerate ground states with edge-spinons in the Haldane phase of spin-111 chain may be reinterpreted as our spin-charge separated edge modes in an equivalent spin-1/21/21/2 fermionic model.