Lattice 2-group symmetries: operators, defects, and gauging

Abstract

We construct and study lattice realizations of finite 2-group symmetries in $2+1$d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group $G$ and 1-form symmetry group $A$: split 2-groups with trivial Postnikov class $[\beta ]\in {\mathcal{H}}^3(G,A)$, and central 2-groups with trivial action $\rho :G\to \text{Aut}(A)$. In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice $A$ 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of $\rho$ and $[\beta ]$ for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.

47 pages + appendices