Higher-Form Anomalies on Lattices

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 $H^{d+2}(B^{p+1}G, U(1))$.

💡 Research Summary

The paper addresses a long‑standing problem in lattice many‑body physics: how to define and detect ’t Hooft anomalies of higher‑form symmetries when the underlying Hilbert space is a tensor‑product of local degrees of freedom. The authors begin by emphasizing that higher‑form symmetry generators are not strictly topological at the microscopic level; they become genuinely topological only after an energetic Gauss‑law constraint is imposed at low energies. This observation motivates a construction that works directly with the symmetry operators themselves, without reference to any particular ground state.

In (2+1) dimensions the authors consider a finite Abelian 1‑form symmetry group (G=\bigoplus_j \mathbb{Z}_{N_j}). For each plaquette (p) of a mesoscopic dual lattice (\hat\Lambda) they introduce a Gauss‑law operator (W(g)_p) implemented by a finite‑depth quantum circuit. The operators satisfy three natural conditions: (i) they realize the group multiplication on each plaquette, (ii) they commute on distinct plaquettes, and (iii) the product over all plaquettes is the identity. When the Gauss‑law constraints (W(g)_p=1) are enforced, the line operators built from products of (W(g)_p) become topological, thereby defining a genuine 1‑form symmetry.

The core of the work is a higher‑form analogue of the Else–Nayak index for 0‑form anomalies. Starting from a 0‑cochain (\epsilon\in C^0(\Lambda,G)) they define a symmetry operator (U(\epsilon)=\prod_p W_p^{\epsilon(p)}). Restricting (U(\epsilon)) to a large disk (R) yields a truncated operator (U_R(\epsilon)) in which the Gauss‑law operators on the boundary of (R) are “cut” (truncated) in a prescribed order. By composing three such truncated operators they construct \