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arXiv:2608.12455

Fermionic Anomalies of Finite Symmetries on Lattices

Ameya Chavda, Ryohei Kobayashi

cond-mat.str-elhep-thquant-phSubmitted Aug 12, 2026• Announced Aug 14, 2026• 0 citations• Revised Sep 26, 2026 (v3)
v1 · Aug 12, 2026v2 · Aug 19, 2026v3 · Sep 26, 2026
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Abstract

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 $\mathbb Z_2^F\to G_f\to G_b$. 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 $(n_2,ν_3)$. For $G_f=G_b\times\mathbb Z_2^F$, we further show that a symmetry with trivial lattice anomaly indices $(n_2,ν_3)$ 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 $H^1(BG_b,\mathbb Z_2)$ anomaly layer in continuum QFT. In particular, for $G_b=\mathbb Z_2$, exact lattice symmetries realize only the even $\mathbb Z_4$ subgroup of the continuum $\mathbb Z_8$ classification. In (2+1)D, we identify three successive anomaly layers of cohomological data $(n_2,n_3,ν_4)$. We show that a nontrivial value of any layer obstructs a symmetric SRE state. For $G_f=G_b\times\mathbb Z_2^F$, it also forbids a symmetric invertible state. We also construct a non-onsite $\mathbb Z_4^F$ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with $c_-\in\mathbb{Z}$, therefore forbids any invertible states compatible with onsite $\mathbb{Z}_4^F$ symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.

Comments
22 pages, 2 figures. Updated discussion reflecting the recent identification of Z4F symmetry in the p+ip state, following arXiv:2609.12042

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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.
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