TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.16.4.089
TI  - Higher-group symmetry in finite gauge theory and stabilizer codes
PY  - 2024/04/03
UR  - https://www.scipost.org/SciPostPhys.16.4.089
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 16
IS  - 4
SP  - 089
A1  - Barkeshli, Maissam
AU  - Chen, Yu-An
AU  - Hsin, Po-Shen
AU  - Kobayashi, Ryohei
AB  - 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 $[O_5] ∈ H^5(BG, U(1))$ 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 the (3+1) D $\mathbb{Z}_2$ toric code.
ER  -

@Article{10.21468/SciPostPhys.16.4.089,
	title={{Higher-group symmetry in finite gauge theory and stabilizer codes}},
	author={Maissam Barkeshli and Yu-An Chen and Po-Shen Hsin and Ryohei Kobayashi},
	journal={SciPost Phys.},
	volume={16},
	pages={089},
	year={2024},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.16.4.089},
	url={https://scipost.org/10.21468/SciPostPhys.16.4.089},
}