Non-invertible and higher-form symmetries in 2+1d lattice gauge theories

TL;DR

This work studies exact generalized symmetries in a 2+1d lattice $\mathbb{Z}_2$ gauge theory coupled to Ising matter, revealing a non-invertible symmetry and a mixed anomaly between a 1-form and a 0-form symmetry. It constructs a lattice condensation operator that generates toric-code ground states and a non-invertible duality operator, and it analyzes a type III anomaly that enforces a Higgs=SPT transition along a diagonal phase boundary. The authors extend the framework to non-invertible SPT phases in 2+1d, including two distinct phases related by a generalized Kennedy–Tasaki transformation, and they connect lattice constructions to continuum fusion 2-categories (2-Rep of a 2-group). They further demonstrate how twisted gauging and half-gauging interfaces capture the duality defects and show how the lattice models realize continuum dualities in the Bhardwaj–Bartsch web, highlighting the interplay between non-invertible symmetries, anomalies, and SPT physics. Overall, the paper provides a concrete lattice realization of rich symmetry structures—non-invertible, higher-form, and anomalous—and develops a robust link to continuum field theories via gauging and KT-type transformations, with potential implications for higher-dimensional generalizations and defect dynamics.

Abstract

We explore exact generalized symmetries in the standard 2+1d lattice $\mathbb{Z}_2$ gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the "Higgs=SPT" proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.

Figures (6)

• Figure 1: The Wilson line operator $W_{v_0,v}$ is a product of $\sigma^z_\ell$ along the blue curve from $v_0 = (i_0,j_0)$ to $v = (i,j)$.
• Figure 2: The condensation operator $\mathsf{C}_{\eta}$ is defined as a sum of the $\mathbb{Z}_2^{(1)}$ 1-form symmetry operators $\eta(\gamma)$ over all possible loops $\gamma$ on a periodic square lattice. The sum includes loops in the trivial homology class (for instance, the first two terms on the right-hand side) as well as loops in non-trivial homology classes (for instance, the third and fourth terms on the right-hand side).
• Figure 3: Upon multiplying by the condensation operator $\mathsf{C}_{\eta}$, we can replace $W_{v_0,v}W_{v_0,v'}$ with adjacent $v$,$v'$ by the link variable $\sigma_{\langle v,v' \rangle}^z$. (Note that the two $\sigma^z_\ell$ operators from the two Wilson lines cancel with each other on the link immediately on the right of $v_0$.)
• Figure 4: Configuration of the membrane operator $U(\widehat{\Sigma})$. Here $\widehat{\Sigma}$ is the red region, and its boundary (represented by the red dashed line) is the curve $\widehat{\gamma}$ on the dual lattice. Those links $\langle v, v'\rangle$ having a non-trivial intersection with $\widehat{\gamma}$, i.e., $(\langle v, v'\rangle, \widehat{\gamma}) = 1$, are shown in blue.
• Figure 5: Phase diagram of \ref{['eq:1-form_hamiltonian']}. The Hamiltonian on the green dashed line has an anomaly, separating the trivial and Higgs phases.