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Higher-group symmetry in finite gauge theory and stabilizer codes
by Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, Ryohei Kobayashi
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Submission summary
| Authors (as registered SciPost users): | Yu-An Chen · Po-Shen Hsin · Ryohei Kobayashi |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202307_00028v1 (pdf) |
| Date submitted: | 2023-07-21 20:56 |
| Submitted by: | Kobayashi, Ryohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior work. We also show how the $d$-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the $d$-group symmetry, such as a Controlled-Z gate in (3+1)D $\mathbb{Z}_2$ toric code.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-1-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202307_00028v1, delivered 2024-01-09, doi: 10.21468/SciPost.Report.8388
Strengths
1. Clearly written and systematic.
2. Many concrete examples are presented.
Report
The paper studies the invertible symmetries of Dijkgraaf-Witten gauge theories in all dimensions. It is clearly written with numerous examples presented to illustrate the conclusions. I recommend the paper wholeheartedly for publication with minimal changes. My suggestions pertain only to setting the results in a mathematical context.
- My understanding is that Dijgraaf-Witten theory for a finite group $G$ in D = d+1 dimensions should have $(D-1)$-fusion category symmetry (in the bosonic case and putting aside issues of unitarity) $\Sigma Z((D-2)Vect_G^{\omega_D})$ where $Z$ denote the Drinfeld center and \Sigma condensation completion of the de-looping. Then from a mathematical perspective, I expect the authors are computing the invertible part of this symmetry category. Is that correct?
- Following from the above, I expect general genuine codimension-2 defects (genuine = not attached to a codimension-1 topological defect) are labelled by a conjugacy class in $G$ and a projective $(D-2)$-representation of the centraliser with projective $(D-1)$-cocyle given by the transgression of $\omega_D$. This seems compatible with the discussion at the beginning of section 2, and includes magnetic defects stacked with gauged SPT phases for the unbroken subgroup. However, it may be nice to relate the discussion in 2,2.1 to this mathematical classification. (I appreciate starting with 2.2 the author's consider instead twisted sector magnetic defects attached to codimension-1 topological defects.)
Requested changes
I suggest some discussion of how the results relate to the aforementioned mathematical framework. However, as this is primarily a physics oriented journal, this should be taken as a suggestion only!
Report #1 by Anonymous (Referee 1) on 2023-9-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202307_00028v1, delivered 2023-09-22, doi: 10.21468/SciPost.Report.7851
Strengths
-detailed study of higher-group symmetry in finite gauge theories and several exciting applications, such as the classification of fermionic SPTs and fault-tolerant gates
-very well written and clear
Weaknesses
-outlook with possible future directions could be mentioned
Report
This paper comprehensively studies the higher group symmetry and its 't Hooft anomaly in finite gauge theories. It describes the higher-group symmetry as arising from a generalization of the Witten effect and the charge-flux attachment. It provides several explicit examples, including field theories and lattice models. The applications discussed in the paper are exciting. For instance, studying the fault-tolerant logical gates from this perspective could lead to insights for practical applications.
The paper's presentation and results are excellent. I wholeheartedly recommend publishing it in SciPost Physics.