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arXiv:2507.21267 · Physical Review Letters

Anomaly-Free Symmetries with Obstructions to Gauging and Onsiteability

Wilbur E. Shirley, Carolyn Zhang, Wenjie Ji, Michael Levin

Published Jan 20, 2026• 2 citations
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Abstract

We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.

DOI
10.1103/zrs1-j2xd

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We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.
We clarify the lore that anomaly-free symmetries are either on-site or can be transformed into on-site symmetries. We prove that any finite, internal, anomaly-free symmetry in a 1+1D lattice Hamiltonian system can be disentangled into an on-site symmetry by introducing ancillae and applying conjugation via a finite-depth quantum circuit. We provide an explicit construction of the disentangling circuit using Gauss's law operators and emphasize the necessity of adding ancillae. Our result establishes the converse to a generalized Lieb-Schultz-Mattis theorem by demonstrating that any anomaly-free symmetry admits a trivially gapped Hamiltonian.
We develop a rigorous topological theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states. We also construct ΩΩΩ-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.
An internal symmetry in a lattice model is said to be onsiteable if it can be disentangled into an on-site action by introducing ancillas and conjugating with a finite-depth circuit. A standard lore holds that onsiteability is equivalent to being anomaly-free, which is indeed valid for finite 0-form symmetries in (1+1)D. However, for higher-form symmetries, these notions become inequivalent: a symmetry may be on site while still anomalous. In this Letter, we clarify the conditions for onsiteability of higher-form symmetries by proposing an equivalence between onsiteability and the possibility of higher gauging. For a finite 1-form symmetry in (2+1)D, we show that the symmetry is onsiteable if and only if its 't Hooft anomaly satisfies a specific algebraic condition that ensures the symmetry can be 1-gauged. We further demonstrate that onsiteable 1-form symmetry in (2+1)D can always be brought into transversal Pauli operators by ancillas and circuit conjugation. In generic dimensions, we derive necessary conditions for onsiteability using lattice 't Hooft anomaly of higher-form symmetry, and conjecture a general equivalence between onsiteability and possibility of higher gauging on lattices. We also identify new "lattice anomalies" intrinsic to higher-form symmetry of lattice models, obstructing on-site realization of symmetry beyond the possibility of higher gauging in continuum.
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 invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic Z4F\mathbb Z_4^FZ4F​ symmetry in (2+1) dimensions. For an onsite Z4F\mathbb Z_4^FZ4F​ symmetry, symmetric invertible phases have integer chiral central charge c−∈Zc_-\in\mathbb Zc−​∈Z, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local Z4F\mathbb Z_4^FZ4F​ symmetry of a p+ipp+ipp+ip superconductor with c−=1/2c_-=1/2c−​=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c−∈Zc_-\in \mathbb{Z}c−​∈Z invertible states. The allowed invertible phases are consequently shifted to c−∈Z+1/2c_-\in\mathbb Z+1/2c−​∈Z+1/2. By gauging the fermion parity of the p+ipp+ipp+ip superconductor, one obtains an Ising topological order enriched by a Z4\mathbb Z_4Z4​ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ipp+ipp+ip state admits an enlarged non-onsite U(1)fU(1)^fU(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)fU(1)^fU(1)f symmetry again forbids Chern insulators carrying c−∈Zc_-\in\mathbb{Z}c−​∈Z, while being compatible with invertible states with c−∈Z+1/2c_-\in\mathbb{Z}+1/2c−​∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
We study ’t Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the “anomaly cocycle” using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group H^3(G,U(1)) × H^2(G,U(1)) H 3 ( G , U ( 1 ) ) × H 2 ( G , U ( 1 ) ) . The first factor captures the anomaly in the internal symmetry group G, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving G and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.