Relative anomalies in (2+1)D symmetry enriched topological states

Barkeshli, Maissam; Cheng, Meng

doi:10.21468/scipostphys.8.2.028

Cited by 39 publications

(69 citation statements)

References 46 publications

(101 reference statements)

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“…Such a mathematical construction was briefly outlined in Ref. [37], but has not yet been fully developed. We note that Ref.…”

Section: A Relation To Prior Workmentioning

confidence: 99%

“…Aside from these special cases, completely general results have so far been presented for relative anomalies, for bosonic topological phases [37,46]. Specifically, given a UMTC and choice of how symmetry permutes the anyons, the difference between symmetry fractionalization patterns is classified by the second group cohomology H 2 ρ (G, A) [6].…”

Section: A Relation To Prior Workmentioning

confidence: 99%

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Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Bulmash

Barkeshli

2020

Phys. Rev. Research

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Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a G-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D G-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H 4 (G, U (1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date.

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“…Such a mathematical construction was briefly outlined in Ref. [37], but has not yet been fully developed. We note that Ref.…”

Section: A Relation To Prior Workmentioning

confidence: 99%

Section: A Relation To Prior Workmentioning

confidence: 99%

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Bulmash

Barkeshli

2020

Phys. Rev. Research

Self Cite

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“…In a unique situation, when the 1-dimensional higher bulk is known to be 3d and bosonic, there is no need to compute the center because this bulk is uniquely determined by a 4-cocycle in H 4 (G, U(1)) according to [61]. In this case, one can treat the 4-cocycle directly as the anomaly as in [3,5,75].…”

Section: Jhep09(2020)093mentioning

confidence: 99%

Classification of topological phases with finite internal symmetries in all dimensions

Kong

Lan

Wen

et al. 2020

J. High Energ. Phys.

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We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

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“…2 The data of modular tensor category are characterized by fusions and braidings of the anyons, which obey stringent constraints such as the pentagon and the hexagon identities. The symmetry fractionalization in (2+1)d TQFT has been systematically studied in [18][19][20][21][22][23]. Applying the Lorentz symmetry fractionalization to a non-spin TQFT, the fractionalization map F produces another non-spin TQFT where the spins of some anyons are shifted by 1 2 , while the other TQFT data (such as the fusion algebra and the Hopf braiding of anyons) remain invariant.…”

Section: Introductionmentioning

confidence: 99%

Lorentz symmetry fractionalization and dualities in (2+1)d

Hsin

Shao

2020

SciPost Phys.

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We discuss symmetry fractionalization of the Lorentz group in (2+1)d non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous Z 2 one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.1 See [9,[11][12][13] for related works in (3+1)d.

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Relative anomalies in (2+1)D symmetry enriched topological states

Cited by 39 publications

References 46 publications

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Classification of topological phases with finite internal symmetries in all dimensions

Lorentz symmetry fractionalization and dualities in (2+1)d

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