Fermionic Anomalies of Finite Symmetries on Lattices

Abstract:We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in (1+1)D and (2+1)D, 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 Hilbert spaces, 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 (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data $(n_2,\nu_3)$. For $G_f=G_b\times\mathbb Z_2^F$, we show that a symmetry with trivial anomaly indices $(n_2,\nu_3)$ is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum 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 $(n_2,n_3,\nu_4)$. 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 find that the lattice obstruction to invertible states does not generally coincide with the continuum 't Hooft anomaly. We explicitly construct a $\mathbb Z_4^F$ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states, even though its continuum anomaly is trivial. Our results highlight a mismatch between lattice and continuum fermionic anomalies and motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.

21 pages, 2 figures. Added refs. Strengthened the dynamical constraints from the Z4F symmetry in Section 7

Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)

arXiv:2608.12455 [cond-mat.str-el]

(or arXiv:2608.12455v2 [cond-mat.str-el] for this version)

https://doi.org/10.48550/arXiv.2608.12455 Focus to learn more arXiv-issued DOI via DataCite

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