Mirror anomaly in fermionic topological orders

Mao, Bin-Bin; Wang, Chenjie

doi:10.1103/physrevresearch.2.023339

Cited by 4 publications

(9 citation statements)

References 64 publications

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“…The indicator η 1f describes pure mirror anomaly, which was discussed in Ref. [32], and η 2f , η 3f describe mixed anomalies between M and U f (1). We remark again that the time-reversal counterparts of the expressions in (3) have already been proposed in Ref.…”

Section: Symmetrysupporting

confidence: 54%

“…Anomaly indicators for the DIII class was studied in Ref. 32 and we list it here for completeness. The number n in the forth column is an integer.…”

Section: Symmetrymentioning

confidence: 99%

“…One may consult Ref. 32 for some properties after gauging the fermion parity. Finally, we comment that the relation (12) does not hold in fermionic topological orders.…”

Section: Topological Ordersmentioning

confidence: 99%

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Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Ning,

Mao,

et al. 2021

Preprint

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We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of U (1), SU (2) or SO(3). In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimension reduction and folding approaches, we re-derive the classification of bulk SPT phases and define a complete set of bulk topological invariants for every symmetry group under consideration, and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as anomaly indicators for the surface SETs which carry 't Hooft anomalies of the associated symmetries whenever the bulk is topologically non-trivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the 't Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent. The anomaly indicators obtained in this work can be straightforwardly translated to their timereversal counterparts that apply to the usual topological insulators and topological superconductors, due to a known correspondence between mirror and time-reversal topological phases.

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Section: Symmetrysupporting

confidence: 54%

“…Anomaly indicators for the DIII class was studied in Ref. 32 and we list it here for completeness. The number n in the forth column is an integer.…”

Section: Symmetrymentioning

confidence: 99%

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Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Ning,

Mao,

et al. 2021

Preprint

Self Cite

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“…On the left-hand side (LHS) of Eq. (A2), we obtain LHS ∝ U 34 (013, 123, 0123) U 34 (023, 32 012, 0123)η 012 (23,34) U 45 (014, 124, 0124) U 45 (024, 42 012, 0124)η 012 (24,45) U 45 (024, 234, 0234) U 45 (034, 43 023, 0234)η 023 (34,45) .…”

mentioning

confidence: 80%

“…[39]. A number of subsequent works further studied and rederived these anomaly formulas through various other methods [40][41][42][43].…”

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

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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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Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries

Ning

Mao

et al. 2021

Phys. Rev. B

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Mirror anomaly in fermionic topological orders

Cited by 4 publications

References 64 publications

Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

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

Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries

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