Lattice 2-group symmetries: operators, defects, and gauging
Abstract
We construct and study lattice realizations of finite 2-group symmetries in 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 G and 1-form symmetry group A: split 2-groups with trivial Postnikov class [β]∈ℋ3(G,A), and central 2-groups with trivial action 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 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.
Related
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BibTeX
@misc{brito2026lattice,
title = {{Lattice 2-group symmetries: operators, defects, and gauging}},
author = {Lucas Z. Brito and Salvatore D. Pace},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10682},
eprint = {2609.10682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10682},
}
RIS
TY - UNPB
TI - Lattice 2-group symmetries: operators, defects, and gauging
AU - Lucas Z. Brito
AU - Salvatore D. Pace
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10682
UR - https://arxiv.org/abs/2609.10682
AB - We construct and study lattice realizations of finite 2-group symmetries in 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 G and 1-form symmetry group A: split 2-groups with trivial Postnikov class [β]∈ℋ^3(G,A), and central 2-groups with trivial action 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 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.
ER -