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.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:1303.2194Physical Review BAug 19, 201360 citations

Classification of Gapped Topological Phases in 1D Interacting System

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Bulletin of the American Physical SocietyMar 21, 20110 citations

Topological phases in two-dimensional systems

Emmanuel V. C. Lopes

Jul 18, 20250 citations

Anyonic symmetries and topological defects in Abelian topological phases: An application to the ADE classification

Mayukh Nilay Khan, Jeffrey C. Y. Teo, Taylor L. Hughes

We study symmetries and defects of a wide class of two-dimensional Abelian topological phases characterized by Lie algebras. We formulate the symmetry group of all Abelian topological field theories. The symmetries relabel quasiparticles (or anyons) but leave exchange and braiding statistics unchanged. Within the class of $ADE$ phases, in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one-dimensional gapped phases along the interface between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall (QSH) states. We show that the edge states of the $ADE$ QSH systems can be gapped even in the presence of time-reversal and charge-conservation symmetries. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior.We study symmetries and defects of a wide class of two-dimensional Abelian topological phases characterized by Lie algebras. We formulate the symmetry group of all Abelian topological field theories. The symmetries relabel quasiparticles (or anyons) but leave exchange and braiding statistics unchanged. Within the class of $ADE$ phases, in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one-dimensional gapped phases along the interface between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall (QSH) states. We show that the edge states of the $ADE$ QSH systems can be gapped even in the presence of time-reversal and charge-conservation symmetries. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior.

arXiv:1403.6478Physical Review BDec 30, 201436 citations

Emergence and construction of novel gapless topological phases

Raditya Weda Bomantara

Jan 1, 20170 citations

Gapped symmetric edges of symmetry-protected topological phases

Yuan-Ming Lu, D. H. Lee

Symmetry-protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topological defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between a topological Ising paramagnet and the double-semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground-state degeneracy on a torus.Symmetry-protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topological defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between a topological Ising paramagnet and the double-semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground-state degeneracy on a torus.

arXiv:1311.1807Physical Review BMay 19, 201422 citations

ADE

phases, in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one-dimensional gapped phases along the interface between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall (QSH) states. We show that the edge states of the

ADE

QSH systems can be gapped even in the presence of time-reversal and charge-conservation symmetries. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior.