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

TL;DR

This paper investigates how lattice higher-form symmetries behave at low energies, revealing that non-compact symmetries often remain non-topological, unlike their compact counterparts which tend to become topological, with concrete models and effective theories analyzed.

Contribution

The study provides the first detailed analysis of the flow of lattice higher-form symmetries to topological or non-topological forms at low energies, including explicit models and general theoretical arguments.

Findings

Non-compact higher-form symmetries often remain non-topological at low energies.

Compact higher-form symmetries tend to become topological unless fine-tuned.

Concrete lattice models demonstrate the different behaviors of these symmetries.

Abstract

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 $\mathbb{R}$ generalization of the Kitaev honeycomb model featuring an $\mathbb{R}$ lattice 1-form symmetry. We show that its low-energy effective field theory is a gapless, non-relativistic theory with a non-topological $\mathbb{R}$ 1-form symmetry. In both the lattice model and the effective field theory, we demonstrate that the non-topological $\mathbb{R}$ 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 $\mathbb{Z}_2$ 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 $\mathbb{R}$ and $\mathbb{Z}$ 1-form symmetries) in lattice systems generically remain non-topological at low energies, whereas compact higher-form symmetries (such as $\mathbb{Z}_{n}$ and $U(1)$ 1-form symmetries) generically become topological.