Classification of 
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 Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

Lan, Tian; Kong, Liang; Wen, Xiao-Gang

doi:10.1103/physrevx.8.021074

Cited by 112 publications

(140 citation statements)

References 76 publications

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“…Up to invertible topological orders, Fermionic/bosonic topological orders with symmetry will be thoroughly studied in an upcoming paper Ref. [73]. In this paper, we concentrate on 2+1D fermionic topological orders without symmetry, which are the simplest examples of nondegenerate UBFC's over a SFC.…”

Section: E Classify Topological Orders With Symmetrymentioning

confidence: 99%

“…The 16 modular extensions of F 0 does not form a group under the stacking product because they are not invertible. However, they do form a Z 16 group if we carefully define the stacking F 0 for modular extensions [73,76]. Moreover, the Witt classes of these 16 modular extensions of F 0 also form a Z 16 subgroup of the bosonic Witt group W [84].…”

Section: A Quantization Of Chiral Central Charge Cmentioning

confidence: 99%

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Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

2016

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We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E = sRep(Z f 2 ) describing particles in a fermionic product state without symmetry, ordescribing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p + i p fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) and quantum dimension d = 1 + √ 2.

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Section: E Classify Topological Orders With Symmetrymentioning

confidence: 99%

Section: A Quantization Of Chiral Central Charge Cmentioning

confidence: 99%

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

2016

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“…The classification of SPT and SET orders are given in Ref. 47,48,52,60,[62][63][64][65][66][67], via group cohomology theory, cobordism theory, and (higher) category theory. The on-site symmetry G is also called global symmetry.…”

Section: Symmetry and Higher Symmetry On Latticementioning

confidence: 99%

Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems

Wen

2019

Phys. Rev. B

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IX. Lattice models that realize higher SPT phasessystematic constructions 21 A. Models realizing bosonic SPT phases 21 B. Models realizing bosonic higher SPT phases 21 C. More general bosonic hSPT phases 22 D. Fermionic hSPT phases 23 E. The usefulness of hSPT phases 24 X. Lattice models that realize topological orders with higher symmetry 24 A. The first type of constructions 24 B. The second type of constructions 24 XI. EM condensed matter systems and their higher symmetry 25 A. Space-time complex, cochains, and cocycles 27 B. Path integral and Hamiltonian 30 References 31 arXiv:1812.02517v5 [cond-mat.str-el] 21 Apr 2019 R/Z J− a R/Z J ) e i 2π B 3 I≤J k IJ a R/Z I ( da R/Z J − da R/Z J )− da R/Z I a R/Z J e − B 3 I R/Z J − da R/Z J )− da R/Z I a R/Z J = e i 2π M 4 k IJ ( da R/Z I − da R/Z I )( da R/Z J − da R/Z J R/Z I )( da R/Z J − da R/Z J ) = e − i 2π M 4 I,J K IJ a R/Z I d da R/Z J , (81) where d da R/Z J can be viewed as the Poincaré dual of the worldline of the U (1) monoples. This implies that the effect of the topological term e i 2π M 4 I≤J k IJ ( da R/Z I − da R/Z I )( da R/Z J − da R/Z J

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“…The answer comes to us as one of a class of TQFTs, called Z/2-gauge-gravity theories; these TQFTs are slight generalizations of Dijkgraaf-Witten theories [DW90,FQ93], in which Stiefel-Whitney classes of the underlying manifold can enter the Lagrangian action. Theories of this sort have also been considered by Kapustin [Kap14a,Kap14b], Wen [Wen15,Wen17], and Lan-Kong-Wen [LKW18], though not in this generality.…”

Section: Gauge-gravity Tqftsmentioning

confidence: 99%

The Low-Energy TQFT of the Generalized Double Semion Model

Debray

2019

Commun. Math. Phys.

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The generalized double semion (GDS) model, introduced by Freedman and Hastings, is a lattice system similar to the toric code, with a gapped Hamiltonian whose definition depends on a triangulation of the ambient manifold M , but whose space of ground states does not depend on the triangulation, but only on the underlying manifold. In this paper, we use topological quantum field theory (TQFT) to investigate the low-energy limit of the GDS model. We define and study a functorial TQFT Z GDS in every dimension n such that for every closed (n − 1)-manifold M , Z GDS (M ) is isomorphic to the space of ground states of the GDS model on M ; the isomorphism can be chosen to intertwine the actions of the mapping class group of M that arise on both sides. Throughout this paper, we compare our constructions and results with their known analogues for the toric code.

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Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

Cited by 112 publications

References 76 publications

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems

The Low-Energy TQFT of the Generalized Double Semion Model

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