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

TL;DR

The paper investigates when lattice higher-form symmetries flow to topological higher-form symmetries at low energies. It constructs a solvable $\mathbb{R}$ lattice 1-form symmetry model on a honeycomb lattice (an $\mathbb{R}$ Kitaev model) that yields a gapless, non-relativistic EFT with a non-topological $\mathbb{R}$ 1-form symmetry, and shows that weak local perturbations lift the infinite degeneracy, preventing a flow to topology. In contrast, it analyzes various $\mathbb{Z}_2$ lattice 1-form symmetry variants (toric code and its modifications) and demonstrates that compact lattice 1-form symmetries generically flow to topological 1-form symmetries at low energies, yielding robust ground-state degeneracies, though fine-tuned cases can remain non-topological. A central result is a necessary condition, expressed as $\lim_{L\to\infty}\min_{\mathfrak{s}} \Delta(\mathfrak{s})/N(\mathfrak{s})>0$, that distinguishes topological flow from non-topological behavior, with non-compact symmetries failing this criterion in generic lattice realizations. The work clarifies the role of compactness in lattice higher-form symmetries and offers a framework for constructing low-energy EFTs in the presence of multiple superselection sectors, advancing the understanding of when lattice systems can realize robust topological order via symmetry considerations.

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.

Figures (6)

• Figure 1: Three different types of links in honeycomb lattice are labelled by $X,Y$ and $Z$. Red arrows ${\bf T}_1$ and ${\bf T}_2$ are lattice vectors. Two sublattices are labelled by $A$ and $B$.
• Figure 2: When the $Q_l$ operators from 3 adjacent elementary loops (i.e., hexagons) are added together, the operators in the interior all cancel, and the result is an operator supported entirely in the exterior, which can be viewed as a larger loop. It is this condition that allows us to view the supports of the symmetry operators as closed loops.
• Figure 3: Non-contractible loops on the honeycomb lattice, denoted by $\gamma$ (red loop) and $\eta$ (green loop), respectively.
• Figure 4: Non-contractible loop operators of toric code supported on the lattice and the dual lattice.
• Figure 5: Schematic energy spectrum of a lattice system with a higher-form symmetry that flows to a topological higher-form symmetry at low energies, and satisfies Eq. \ref{['eq: topological condition']}. The horizontal axis represents symmetry sectors characterized by the collection of eigenvalues under all symmetry operators supported on contractible loops. Different sectors are labeled by different colors. Each short line represents an energy eigenstate. $\Delta(\mathfrak{s})$ is the energy difference between the lowest-energy state in sector $\mathfrak{s}$ and the true ground state, $\Delta \equiv \min_\mathfrak{s}\{\Delta(\mathfrak{s})\}$, and $N(\mathfrak{s})$ is the number of contractible loop eigenvalues differing from those in the ground-state sector. The spectrum in the ground-state symmetry sector (labeled red) can be either gapped or gapless.