Higher-group symmetry in finite gauge theory and stabilizer codes

TL;DR

The paper develops a comprehensive framework for higher-group symmetry in finite gauge theories by constructing a $d$-group symmetry in $(d+1)$D and deriving its 't Hooft anomaly. A generalized Witten effect and charge-flux attachment show how 1-form and higher-form symmetries mix nontrivially with gauged SPT defects, yielding non-Abelian fusion and richer defect structures even when magnetic fluxes are center holonomies. The authors apply these ideas to bosonic and fermionic theories, including Dijkgraaf-Witten twists, and demonstrate lattice realizations that reveal the interplay between 0-, 1-, and 2-form symmetries, with consequences for Clifford hierarchy and fault-tolerant logical gates in stabilizer codes. They provide explicit examples in $\mathbb{Z}_2$, $\mathbb{Z}_2^2$, and $\mathbb{Z}_2^3$ gauge theories, derive simpler obstructions for fermionic SPT classifications, and connect higher-group data to stabilizer code gates such as a controlled-Z in the (3+1)D toric code, highlighting both foundational and practical implications for topological phases and quantum computation.

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.

Figures (5)

• Figure 1: The magnetic defect is attached to a gauged $G$ SPT defect (that supports $G$ gauge theory with a topological action) in the presence of topological action, with the gauged SPT defect given by integrating the topological action over the circle fiber with given holonomy of the magnetic defect. The magnetic and gauged SPT defects both extend in the remaining transverse $(D-2)$ dimensions, which we suppress in the figure.
• Figure 2: Junction of magnetic defects can emit a gauged SPT defect of dimension $(D-2)$.
• Figure 3: Junction of 0-form symmetry defects indicated by black and dashed lines (with red arrows indicating their orientations to specify the action of the symmetry). On the right, we include a magnetic flux loop encircling the junction, and there are semion and anti-semion particles at the intersection of the flux loop with the non-trivial domain walls, indicated by the green and orange points. To compensate for their difference, there is an electric charge at the junction, indicated by the red dot.
• Figure 5: The configurations of the operators $U(R), V(\Sigma), W(\gamma)$ that appear in the commutation relation Eq. \ref{['eqn:threegroupcommutationZ2']}.
• Figure 6: When the magnetic surface has a non-trivial trivalent junction, it is regarded as a fusion of two magnetic flux loops. The magnetic flux at the domain wall is dressed with electric charge 1/2, which is represented by an orange dot. When the junction crosses through the codimension-1 defect, it leaves an electric charge on the defect due to the ${\mathbb Z}_4$ fusion rule of the magnetic flux on the defect. A similar figure can also be found in Hsin:2019fhf.