Higher-group symmetry in finite gauge theory and stabilizer codes

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 dd-group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1)(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))[O5]∈H5(BG,U(1)) obstruction that has appeared in prior work. We also show how the dd-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 dd-group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2ℤ2 toric code.