Higher-Form Anomalies on Lattices

TL;DR

This work develops an operator-based framework to diagnose ’t Hooft anomalies of higher-form symmetries in lattice models with tensor-product Hilbert spaces, where symmetry generators become topological only after enforcing a Gauss-law constraint. In (2+1)D, it introduces an Else–Nayak–style index valued in $H^4(B^2G, U(1))$ for 1-form symmetries and proves its invariance under phase and local-operator redefinitions, with nonzero values signaling obstructions to symmetric short-range entangled states. The framework is extended to arbitrary $(d+1)$D spacetime dimensions, where the anomaly is captured by a cohomology class in $H^{d+2}(B^{p+1}G, U(1))$ through a dimension-reduction construction that successively lowers the form degree. A concrete ${ m Z}_2$ 1-form symmetry example in (2+1)D yields a nontrivial $oldsymbol{ ho}_4$ corresponding to the $(3+1)$D response $S i rac{oldsymbol{ ho}_4}{2} abla B ty B$, illustrating a direct lattice-to-continuum connection. Overall, the paper provides a unified, state-independent approach to higher-form anomalies on lattices and clarifies links to known invariants such as T-junction contributions and Pontryagin squares, with implications for higher-group symmetries and lattice formulations of topological phases.

Abstract

Higher-form symmetry in a tensor product Hilbert space is always emergent: the symmetry generators become genuinely topological only when the Gauss law is energetically enforced at low energies. In this paper, we present a general method for defining the 't Hooft anomaly of higher-form symmetries in lattice models built on a tensor product Hilbert space. In (2+1)D, for given Gauss law operators realized by finite-depth circuits that generate a finite 1-form $G$ symmetry, we construct an index representing a cohomology class in $H^4(B^2G, U(1))$, which characterizes the corresponding 't Hooft anomaly. This construction generalizes the Else-Nayak characterization of 0-form symmetry anomalies. More broadly, under the assumption of a specified formulation of the $p$-form $G$ symmetry action and Hilbert space structure in arbitrary $d$ spatial dimensions, we show how to characterize the 't Hooft anomaly of the symmetry action by an index valued in $H^{d+2}(B^{p+1}G, U(1))$.

Figures (7)

• Figure 1: (a): The symmetry operators are supported at the thickened dual lattice of a mesoscopic triangulation of a 2d lattice system. $W_p$ is supported at a closed loop along the boundary of a plaquette $p$. A plaquette of the dual lattice $p$ corresponds to a vertex in the original lattice $\Lambda$. (b): The local operator $O_e$ is supported at the intersection between an edge $e$ of $\partial R$ and an edge of $\hat{\Lambda}$.
• Figure 2: An interval $I$ at the boundary of the region $R$.
• Figure 3: The 0-forms $\epsilon_{ij}$, group elements $g_{ijk}$ are associated with the 1-simplices, 2-simplices of a simplicial complex. The operators $\Omega,A$, and $\omega$ are associated with 2,3,4-simplices respectively.
• Figure 4: (a): The truncated symmetry operator $W_j$ of the ${\mathbb Z}_2$ toric code. A single qubit is located at each edge of a square lattice, and the operator is a product of Pauli $X$ and $Z$ operators. (b): At the boundary of the region $R$, there is an array of truncated operators $\ldots, W_{j-1}, W_j, W_{j+1}, \ldots$ The neighboring operators anti-commute with each other due to fermionic statistics.
• Figure 5: The 0-forms $\epsilon_{ij}$, group elements $g_{ijk}$ are associated with the 1-simplices, 2-simplices of a simplicial complex. The state label $a$ is associated with a 0-simplex. The $j$-form $A_j$ is associated with a $(d+1-j)$-simplex.