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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
Submission summary
| Authors (as registered SciPost users): | Yu-An Chen · Po-Shen Hsin · Ryohei Kobayashi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2211.11764v3 (pdf) |
| Date accepted: | March 12, 2024 |
| Date submitted: | March 7, 2024, 7:28 a.m. |
| Submitted by: | Po-Shen Hsin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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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.
Author comments upon resubmission
List of changes
"On the boundary of the (D − 1)-dimensional gauged SPT defect, there is a projective (D − 2)-representation of the centraliser with projective (D − 1)-cocyle given by the transgression of the D-dimensional topological action."
-p16: We add footnote 16 as suggested by referee report 2:
"The Dijkgraaf-Witten theory of finite group G and topological action ω(D) has (D − 1)-fusion category symmetry.
Mathematically, the above symmetries describe its invertible parts."
Published as SciPost Phys. 16, 089 (2024)