Topological invariants for gauge theories and symmetry-protected topological phases

Wang, Chenjie; Levin, Michael

doi:10.1103/physrevb.91.165119

Cited by 100 publications

(166 citation statements)

References 53 publications

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“…As we see, the presence of fermionic particles indeed enables new types of three-loop braiding statistics, forbidden when the particles are bosonic. Because of the well-established correspondence between SPT phases and gauge theories [2,7,[22][23][24], our results also imply existence of intrinsic FSPT phases. We derive these results through a combination of physical arguments and exactly solvable models.…”

Section: Introductionsupporting

confidence: 70%

“…In such gauged BSPT phases, particle excitations are the gauge charges and loop excitations are vortices carrying gauge fluxes. Most importantly, distinct BSPT phases give rise to different "three-loop" braiding statistics in the gauge theories, which is a fundamentally new type of braiding statistics in 3D [2,3,[7][8][9][10][11][12][13]. In other words, three-loop braiding statistics serves as a topological invariant for BSPT phases.…”

Section: Introductionmentioning

confidence: 99%

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Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

2018

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We study Abelian braiding statistics of loop excitations in three-dimensional gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the presence of fermionic particles, which correspond to 3D "intrinsic" FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with antiunitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is Z 2 × Z 4 . To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.

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Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

2018

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“…This mixed version of three-loop statistics enriches our previous understandings on three-loop statistics among gauge fluxes [26][27][28][29][30]. Pictorially, the topological invariant corresponds to the three-loop process shown in Fig.…”

Section: Seg(z2 Zk ) With K ∈ Zeven Inmentioning

confidence: 69%

“…The coupling term in Eq. (21) is now changed to: (27) which means that both Z N1 and Z N2 symmetry charges are carried by the first layer. We will show that (LCM stands for "least common multiple"):…”

Section: Summary Of the 5-step General Proceduresmentioning

confidence: 99%

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Symmetry enrichment in three-dimensional topological phases

Ning

Liu

2016

Phys. Rev. B

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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.

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From gauge to higher gauge models of topological phases

Delcamp

Tiwari

2018

J. High Energ. Phys.

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We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and 't Hooft anomalies in the context of these higher group symmetries. CONTENTS

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Topological invariants for gauge theories and symmetry-protected topological phases

Cited by 100 publications

References 53 publications

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

Symmetry enrichment in three-dimensional topological phases

From gauge to higher gauge models of topological phases

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