Relative Anomalies in (2+1)D Symmetry Enriched Topological States

Submission summary

 As Contributors: Maissam Barkeshli · Meng Cheng Arxiv Link: https://arxiv.org/abs/1906.10691v2 Date accepted: 2019-12-17 Date submitted: 2019-12-11 Submitted by: Barkeshli, Maissam Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

Abstract

Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for $\mathbb{Z}_2^{\bf T}$ space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for $U(1) \times \mathbb{Z}_2^{\bf T}$ and $U(1) \rtimes \mathbb{Z}_2^{\bf T}$ symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as $\mathbb{Z}_4^{\bf T}$ and mixed anomalies for $\mathbb{Z}_2 \times \mathbb{Z}_2^{\bf T}$ symmetry, and unitary $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry with non-trivial anyon permutations.

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Published as SciPost Phys. 8, 028 (2020)

We thank the referee for his/her careful reading of our manuscript.

Regarding the referee's first question, indeed this was discussed in the paper by Etingof, Nikshych, and Ostrik (see https://arxiv.org/pdf/0909.3140.pdf, Proposition 8.12 ). The formulas derived by ENO apply to the case where the symmetry actions do not permute the anyons. The generalization to the case where anyons are permuted has not been proven, as far as we are aware.

Regarding the referee's second comment, we have included the discussion of the Abelian case summarized by the referee at the end of Sec. VI of our revised version.

Finally, we have made the additional grammatical changes suggested by the referee.

List of changes

1. Added a short discussion on the relation between the relative anomaly formula for Z2^T symmetry for Abelian anyons and the arf invariant discussed in the paper of Lee and Tachikawa (2018) .

2. Fixed a typo in Table I and clarified the theories it applies to.

3. Fixed some minor typos and grammatical issues mentioned by the referee.

Submission & Refereeing History

Resubmission 1906.10691v2 on 11 December 2019
Submission 1906.10691v1 on 19 July 2019