• Corpus ID: 253761101

Higher-group symmetry in finite gauge theory and stabilizer codes

@inproceedings{Barkeshli2022HighergroupSI,
  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},
  year={2022}
}

• M. Barkeshli, Yu-An Chen, +1 author R. Kobayashi
• Published 21 November 2022
• Mathematics

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, 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 than has appeared in previous 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 construct new logical gates in stabilizer codes using the d -group symmetry, such as the control-Z gate in (3+1)D Z 2 toric code.