Higher-group symmetry in finite gauge theory and stabilizer codes

• URL: http://arxiv.org/abs/2211.11764v3
• Date: Wed, 6 Mar 2024 02:10:11 GMT
• Title: Higher-group symmetry in finite gauge theory and stabilizer codes
• Authors: Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, Ryohei Kobayashi
• Abstract summary: A large class of gapped phases of matter can be described by topological finite group gauge theories. We derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1) space-time dimensions. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups.
• Score: 3.8769921482808116
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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