Topological Nematic States and Non-Abelian Lattice Dislocations

Barkeshli, Maissam; Qi, Xiao-Liang

doi:10.1103/physrevx.2.031013

Cited by 257 publications

(416 citation statements)

References 68 publications

Supporting

Mentioning

414

Contrasting

Order By: Relevance

“…Experimentally, it is of interest to detect patterns of such symmetry fractionalization, which may help us characterize QSLs [48]. In addition to the usual global symmetry, there are also SETs enriched by a new kind of symmetry dubbed "topological (anyonic)" symmetry [43,[49][50][51][52][53][54][55][56][57][58][59][60]. This symmetry denotes an automorphism of the topological data (braiding statistics, quantum dimensions, etc.).…”

Section: Introductionmentioning

confidence: 99%

Symmetry enrichment in three-dimensional topological phases

Ning

Liu

2016

Phys. Rev. B

View full text Add to dashboard Cite

While two-dimensional symmetry-enriched topological phases (SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by GT) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on SEGs with gauge group Gg = ZN 1 × ZN 2 × · · · and on-site unitary symmetry group Gs = ZK 1 × ZK 2 × · · · or Gs = U(1) × ZK 1 × · · · . Each SEG(Gg, Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., SET orders) of SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from SEGs to SETs. By giving full dynamics to background gauge fields, SEGs may be eventually promoted to a set of new gauge theories (denoted by GT * ). Based on their gauge groups, GT * s may be further regrouped into different classes each of which is labeled by a gauge group G * g . Finally, a web of gauge theories involving GT, SEG, SET and GT * is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

show abstract

Section: Introductionmentioning

confidence: 99%

Symmetry enrichment in three-dimensional topological phases

Ning

Liu

2016

Phys. Rev. B

View full text Add to dashboard Cite

show abstract

“…Dislocations are of central interest in this endeavor, being the topological defects exclusively related to the lattice translations. In two dimensions (2D), the role of these lattice defects has been recently elucidated in TBIs [10,11], as well as in topological superconductors [12,13] and interacting topological states [14][15][16]. In particular, it has been shown that these lattice defects in two-dimensional TBIs act as probes of distinct topological states through binding of the localized zero-energy modes [10].…”

mentioning

confidence: 99%

Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band insulators

et al. 2014

View full text Add to dashboard Cite

We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this K-b-t rule, the lattice topology, represented by dislocation lines oriented in direction t with Burgers vector b, combines with the electronic-band topology, characterized by the band-inversion momentum K inv , to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux = K inv · b(mod 2π ). Although it has already been discovered by Ran et al. [Nat. Phys. 5, 298 (2009)] that dislocation lines host propagating modes, the exact mechanism of their appearance in conjunction with the crystal symmetries of a topological state is provided by the K-b-t rule. Finally, we discuss possible experimentally consequential examples in which the modes are oblivious to the direction of propagation, such as the recently proposed topologically insulating state in electron-doped BaBiO 3 . Topological band insulators (TBIs) represent a new class of quantum materials that, due to the presence of time-reversal symmetry (TRS), feature an insulating bulk band gap together with metallic edge or surface states protected by a Z 2 topological invariant [1][2][3][4]. This Z 2 classification of TBIs is a part of the classification of free gapped fermion matter in the presence of the fundamental antiunitary time-reversal and particle-hole symmetries, the so-called tenfold way [5][6][7]. On the other hand, topologically insulating crystals break continuous translational and rotational symmetries down to discrete symmetries mathematically characterized by the space groups. By considering the crystal symmetries, an extra layer in this Z 2 classification of TBIs has been recently uncovered [8]. This space group classification of TBIs results in the enrichment of the tenfold way with extra phases, such as crystalline (or "valley") phases [9] and translationally active phases, the latter featuring an odd number of band inversions at non-points in the Brillouin zone (BZ) [8,10]. Dislocations are of central interest in this endeavor, being the topological defects exclusively related to the lattice translations. In two dimensions (2D), the role of these lattice defects has been recently elucidated in TBIs [10,11], as well as in topological superconductors [12,13] and interacting topological states [14][15][16]. In particular, it has been shown that these lattice defects in two-dimensional TBIs act as probes of distinct topological states through binding of the localized zero-energy modes [10].Although early on it was identified that in three-dimensional TBIs dislocation lines support propagating helical modes [17], the precise role of dislocations has not been explored thoroughly [18][19][20]. In particular, the relation between the lattice symmetry and the electronic topology, as well as the characterization of these topological states through the response of the dislocation lines, still needs to be addressed. Dislocations in thr...

show abstract

“…The gapping conditions of Abelian topological orders have recently been understood in terms of the concept of Lagrangian subsets [17][18][19][20][21][22][23][24], and subsequently the GSD of these Abelian phases on open surfaces with multiple boundaries were computed [23,25], based on the idea of anyon transport across boundaries. Experiments detecting and utilizing the topological degeneracy with gapped boundaries were proposed in [26,27]. The gapping conditions of non-Abelian topological orders have recently been tackled by the mechanism of anyon condensation [28] and equivalently by solving certain algebraic equations [29].…”

Section: Jhep01(2018)134 1 Introduction and Summarymentioning

confidence: 99%

Boundary Hamiltonian theory for gapped topological phases on an open surface

Luo

Pankovich

et al. 2018

J. High Energ. Phys.

View full text Add to dashboard Cite

In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.

show abstract

Topological Nematic States and Non-Abelian Lattice Dislocations

Cited by 257 publications

References 68 publications

Symmetry enrichment in three-dimensional topological phases

Symmetry enrichment in three-dimensional topological phases

Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band insulators

Boundary Hamiltonian theory for gapped topological phases on an open surface

Contact Info

Product

Resources

About