Defining 2-group symmetries on lattices and extracting their anomalies

Abstract

Ordinary $0$-form symmetries $G$ can mix with $1$-form symmetries ${\Pi }_2$ to form a $2$-group $\mathcal{G}$. We give a concrete lattice definition for a class of finite $2$-group symmetries in terms of local relations among symmetry operators. These relations identify the Postnikov class $[\rho ]\in H^3(BG,{\Pi }_2)$ that characterizes the mixing. We develop a method to extract anomalies of these symmetries in one spatial dimension by restricting symmetry operations to an interval and comparing their composition near its endpoints. We also construct explicit qubit symmetry actions in $d=1,2,3$ spatial dimensions for a $2$-group with $G={\Pi }_2={\mathbb{Z}}_2$. These actions share the same nontrivial Postnikov class but include both anomaly-free and anomalous realizations. Their cohomological anomalies are classified by $H^{d+2}(B\mathcal{G},U(1))$, which is ${\mathbb{Z}}_2$ for $d=1,2$ and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for $d=3$, and our constructions realize every class in these dimensions. Our results connect higher-group symmetry and its anomalies to concrete lattice operations, providing a microscopic framework for studying anomalous symmetries.

arXiv comment45 pages, 9 figures; Added acknowledgement, 3.4.2 deleted, Appendix C restructured