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arXiv:2507.07430

Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

Kyle Kawagoe, Wilbur E. Shirley

Published Jul 10, 2025• 0 citations
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Abstract

We describe a method for computing the anomaly of any finite unitary symmetry group $G$ acting by finite-depth quantum circuits on a two-dimensional lattice system. The anomaly is characterized by an index valued in the cohomology group $H^4(G,U(1))$, 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 $G=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.

DOI
10.48550/arxiv.2507.07430

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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 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.
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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)).
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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.