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 $H^1(B{G}_b,{\mathbb{Z}}_2)$ 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