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arXiv:2509.12304 · Physical Review Letters

Higher-Form Anomalies on Lattices

Yitao Feng, Ryohei Kobayashi, Yu-An Chen, Shinsei Ryu

Published Jan 5, 2026• 0 citations
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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 Letter, 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^{2}G,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)).

DOI
10.1103/2jz1-m1lb

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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 Letter, 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^{2}G,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)).
We define an 't Hooft anomaly index for a group acting on a 2d quantum lattice system by finite-depth circuits. It takes values in the degree-4 cohomology of the group and is an obstruction to on-siteability of the group action. We introduce a 3-group (modeled as a crossed square) describing higher symmetries of a 2d lattice system and show that the 2d anomaly index is an obstruction for promoting a symmetry action to a morphism of 3-groups. This demonstrates that 't Hooft anomalies are a consequence of a mixing between ordinary symmetries and higher symmetries. Similarly, to any 1d lattice system we attach a 2-group (modeled as a crossed module) and interpret the Nayak-Else anomaly index as an obstruction for promoting a group action to a morphism of 2-groups. The meaning of indices of Symmetry Protected Topological states is also illuminated by higher group symmetry.
Ordinary 000-form symmetries GGG can mix with 111-form symmetries Π2Π_2Π2​ to form a 222-group G\mathcal{G}G. We give a concrete lattice definition for a class of finite 222-group symmetries in terms of local relations among symmetry operators. These relations identify the Postnikov class [ρ]∈H3(BG,Π2)[ρ]\in H^3(BG,Π_2)[ρ]∈H3(BG,Π2​) that characterizes the mixing. We develop a method to extract anomalies of these symmetries in one spatial dimension by restricting symmetry operations to an interval and comparing their composition near its endpoints. We also construct explicit qubit symmetry actions in d=1,2,3d=1,2,3d=1,2,3 spatial dimensions for a 222-group with G=Π2=Z2G=Π_2=\mathbb{Z}_2G=Π2​=Z2​. These actions share the same nontrivial Postnikov class but include both anomaly-free and anomalous realizations. Their cohomological anomalies are classified by Hd+2(BG,U(1))H^{d+2}(B\mathcal G,U(1))Hd+2(BG,U(1)), which is Z2\mathbb{Z}_2Z2​ for d=1,2d=1,2d=1,2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​ for d=3d=3d=3, and our constructions realize every class in these dimensions. Our results connect higher-group symmetry and its anomalies to concrete lattice operations, providing a microscopic framework for studying anomalous symmetries.
We describe a method for computing the anomaly of any finite unitary symmetry group GGG acting by finite-depth quantum circuits on a two-dimensional lattice system. The anomaly is characterized by an index valued in the cohomology group H4(G,U(1))H^4(G,U(1))H4(G,U(1)), which generalizes the Else-Nayak index for locality preserving symmetries of quantum spin chains. We show that a nontrivial index precludes the existence of a trivially gapped symmetric Hamiltonian; it is also an obstruction to ``onsiteability" of the symmetry action. We illustrate our method via a simple example with G=Z2×Z2×Z2×Z2G=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2G=Z2​×Z2​×Z2​×Z2​. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.
We construct and study lattice realizations of finite 2-group symmetries in 2+1{2+1}2+1d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group GGG and 1-form symmetry group AAA: split 2-groups with trivial Postnikov class [β]∈H3(G,A){[β]\in\mathcal{H}^3(G,A)}[β]∈H3(G,A), and central 2-groups with trivial action ρ ⁣:G→Aut(A){ρ\colon G\to\text{Aut}(A)}ρ:G→Aut(A). In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice AAA 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of ρρρ and [β][β][β] for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.
We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in one and two spatial dimensions, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. We consider lattice systems formed by tensor product of onsite fermionic and bosonic local degrees of freedom, and finite internal symmetry given by a central extension Z2F→Gf→Gb\mathbb Z_2^F\to G_f\to G_bZ2F​→Gf​→Gb​. We extract a hierarchy of fermionic anomaly indices for a given symmetry operator in both (1+1)D and (2+1)D. In (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data (n2,ν3)(n_2,ν_3)(n2​,ν3​). For Gf=Gb×Z2FG_f=G_b\times\mathbb Z_2^FGf​=Gb​×Z2F​, we further show that a symmetry with trivial lattice anomaly indices (n2,ν3)(n_2,ν_3)(n2​,ν3​) is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum fermionic QFT, we find that exact lattice symmetries do not realize the additional H1(BGb,Z2)H^1(BG_b,\mathbb Z_2)H1(BGb​,Z2​) anomaly layer in continuum QFT. In particular, for Gb=Z2G_b=\mathbb Z_2Gb​=Z2​, exact lattice symmetries realize only the even Z4\mathbb Z_4Z4​ subgroup of the continuum Z8\mathbb Z_8Z8​ classification. In (2+1)D, we identify three successive anomaly layers of cohomological data (n2,n3,ν4)(n_2,n_3,ν_4)(n2​,n3​,ν4​). We show that a nontrivial value of any layer obstructs a symmetric SRE state. For Gf=Gb×Z2FG_f=G_b\times\mathbb Z_2^FGf​=Gb​×Z2F​, it also forbids a symmetric invertible state. We also construct a non-onsite Z4F\mathbb Z_4^FZ4F​ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with c−∈Zc_-\in\mathbb{Z}c−​∈Z, therefore forbids any invertible states compatible with onsite Z4F\mathbb{Z}_4^FZ4F​ symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.
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 R\mathbb{R}R generalization of the Kitaev honeycomb model featuring an R\mathbb{R}R lattice 1-form symmetry. We show that its low-energy effective field theory is a gapless, non-relativistic theory with a non-topological R\mathbb{R}R 1-form symmetry. In both the lattice model and the effective field theory, we demonstrate that the non-topological R\mathbb{R}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 Z2\mathbb{Z}_2Z2​ 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 R\mathbb{R}R and Z\mathbb{Z}Z 1-form symmetries) in lattice systems generically remain non-topological at low energies, whereas compact higher-form symmetries (such as Zn\mathbb{Z}_{n}Zn​ and U(1)U(1)U(1) 1-form symmetries) generically become topological.