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arXiv:2505.04719

Higher symmetries and anomalies in quantum lattice systems

Anton Kapustin, Xu Shi-xiong

Published May 7, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2505.04719

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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.
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 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.
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)).
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.
't Hooft anomalies impose fundamental constraints on quantum matter and often lead to emergent symmetry structures upon gauging. We analyze a lattice model with four global symmetries realizing a mixed anomaly described by ∼a1∧a2∧a3∧a4\sim a_1\wedge a_2\wedge a_3\wedge a_4∼a1​∧a2​∧a3​∧a4​, where the aia_iai​ denote background gauge fields for the global symmetries. Through explicit lattice gauging, we demonstrate the emergence of higher symmetry structures, including 2-group, non-invertible, and higher fusion categorical symmetries. We also provide a field-theoretical understanding of these results. Applying this framework to systems with Lieb-Schultz-Mattis anomalies, obtained by promoting part of the internal symmetries to translational symmetries, we demonstrate that modulated (dipole) symmetries arise as direct counterparts of those in systems with purely internal typeIV anomalies. Importantly, we uncover a qualitatively new feature absent in previously studied modulated symmetries: their realization can become intrinsically defect-dependent. In particular, the emergent symmetry structure changes depending on whether symmetry defects are present. This work establishes a concrete lattice realization of mixed anomalies and reveals a rich structure of emergent symmetries, thereby clarifying their role in constraining quantum phases of matter.
Abstract We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and ’t Hooft anomalies in the context of these higher group symmetries.