When does a lattice higher-form symmetry flow to a topological higher-form symmetry at low energies?

We study the lattice version of higher-form symmetries on tensor-product Hilbert spaces. Interestingly, at low energies, these symmetries may not flow to the topological higher-form symmetries familiar from relativistic quantum field theories, but instead to non-topological higher-form symmetries. We present concrete lattice models exhibiting this phenomenon. One particular model is an R ℝ generalization of the Kitaev honeycomb model featuring an R ℝ lattice 1-form symmetry. We show that its low-energy effective field theory is a gapless, non-relativistic theory with a non-topological R ℝ 1-form symmetry. In both the lattice model and the effective field theory, we demonstrate that the non-topological R ℝ 1-form symmetry is not robust against local perturbations. In contrast, we also study various modifications of the toric code and their low-energy effective field theories to demonstrate that the compact Z_2 ℤ 2 lattice 1-form symmetry does become topological at low energies unless the Hamiltonian is fine-tuned. Along the way, we clarify the rules for constructing low-energy effective field theories in the presence of multiple superselection sectors. Finally, we argue on general grounds that non-compact higher-form symmetries (such as R ℝ and Z ℤ 1-form symmetries) in lattice systems generically remain non-topological at low energies, whereas compact higher-form symmetries (such as Z_n ℤ n and U(1) U ( 1 ) 1-form symmetries) generically become topological.