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arXiv:2609.10682

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

Lucas Z. Brito, Salvatore D. Pace

cond-mat.str-elhep-thquant-phSubmitted Sep 9, 2026• Announced Sep 11, 2026• 0 citations
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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 ${[β]\in\mathcal{H}^3(G,A)}$, and central 2-groups with trivial action ${ρ\colon G\to\text{Aut}(A)}$. 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 $ρ$ and $[β]$ 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.

Comments
47 pages + appendices

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We construct and study lattice realizations of finite 2-group symmetries in 2+1{2+1}2+1d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group GGG and 1-form symmetry group AAA: split 2-groups with trivial Postnikov class [β]∈H3(G,A){[β]\in\mathcal{H}^3(G,A)}[β]∈H3(G,A), and central 2-groups with trivial action ρ ⁣:G→Aut(A){ρ\colon G\to\text{Aut}(A)}ρ:G→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 AAA 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 ρρρ and [β][β][β] 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.
Higher-form symmetries generalize conventional global symmetries and act on lower-dimensional submanifolds of a quantum system. While Abelian topological phases can be organized by 1-form symmetries that form a group, non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras. In this work, we make this statement precise in quantum double lattice models D(G)\mathcal D(G)D(G) for finite non-Abelian groups GGG. We construct the electric, magnetic, and dyonic 1-form operators directly at the lattice fixed point and show that together they form a complete nonlocal diagnostic algebra for the topological Hilbert space. Using these operators, we explicitly determine the cylinder and torus ground-state subspaces for arbitrary finite GGG. Furthermore, we calculate the microscopic fusion and gluing of the 1-form symmetries and show that their topological deformation properties emerge after projection to the defect-free topological subspace. Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking and provide an operator language that may be useful for characterizing such states in quantum processors.
Ordinary 000-form symmetries GGG can mix with 111-form symmetries Π2Π_2Π2​ to form a 222-group G\mathcal{G}G. We give a concrete lattice definition for a class of finite 222-group symmetries in terms of local relations among symmetry operators. These relations identify the Postnikov class [ρ]∈H3(BG,Π2)[ρ]\in H^3(BG,Π_2)[ρ]∈H3(BG,Π2​) 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=1,2,3d=1,2,3d=1,2,3 spatial dimensions for a 222-group with G=Π2=Z2G=Π_2=\mathbb{Z}_2G=Π2​=Z2​. These actions share the same nontrivial Postnikov class but include both anomaly-free and anomalous realizations. Their cohomological anomalies are classified by Hd+2(BG,U(1))H^{d+2}(B\mathcal G,U(1))Hd+2(BG,U(1)), which is Z2\mathbb{Z}_2Z2​ for d=1,2d=1,2d=1,2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​ for d=3d=3d=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.
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We propose an index of non-invertible symmetry operators in 1+1 dimensions and discuss its relation to the realizability of non-invertible symmetries on the tensor product of finite dimensional on-site Hilbert spaces on the lattice. Our index generalizes the Gross-Nesme-Vogts-Werner index of invertible symmetry operators represented by quantum cellular automata (QCAs). Assuming that all fusion channels of symmetry operators have the same index, we show that the fusion rules of finitely many symmetry operators on a tensor product Hilbert space can agree, up to QCAs, only with those of weakly integral fusion categories. We also discuss an attempt to establish an index theory for non-invertible symmetries within the framework of tensor networks. To this end, we first propose a general class of matrix product operators (MPOs) that describe non-invertible symmetries on a tensor product Hilbert space. These MPOs, which we refer to as topological injective MPOs, include all invertible symmetries, non-anomalous fusion category symmetries, and the Kramers-Wannier symmetries for finite abelian groups. For topological injective MPOs, we construct the defect Hilbert spaces and the corresponding sequential quantum circuit representations. We also show that all fusion channels of topological injective MPOs have the same index if there exist fusion and splitting tensors that satisfy appropriate conditions. The existence of such fusion and splitting tensors has not been proven in general, although we construct them explicitly for all examples of topological injective MPOs listed above.
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