Fermionic Anomalies of Finite Symmetries on Lattices
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 ℤ2F→ 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). For Gf=Gb×ℤ2F, we further show that a symmetry with trivial lattice anomaly indices (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) anomaly layer in continuum QFT. In particular, for Gb=ℤ2, exact lattice symmetries realize only the even ℤ4 subgroup of the continuum ℤ8 classification. In (2+1)D, we identify three successive anomaly layers of cohomological data (n2,n3,ν4). We show that a nontrivial value of any layer obstructs a symmetric SRE state. For Gf=Gb×ℤ2F, it also forbids a symmetric invertible state. We also construct a non-onsite ℤ4F lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with c-∈ℤ, therefore forbids any invertible states compatible with onsite ℤ4F symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.
Related
- Defining 2-group symmetries on lattices and extracting their anomalies
- Generalised Symmetries, Anomalies, and Maximal Branches of 3d Chern-Simons Matter Theories
- Dissipative Majorana Chain---Anomalies and Non-Invertible Symmetries
- Quantum Dynamics of Full Counting Statistics in Fermionic Lattices with Localized Gain
- Anomalies, higher symmetries, and homotopy theory I: crossed precosheaves and a nonabelian Cech construction
BibTeX
@misc{chavda2026fermionic,
title = {{Fermionic Anomalies of Finite Symmetries on Lattices}},
author = {Ameya Chavda and Ryohei Kobayashi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12455},
eprint = {2608.12455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12455},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fermionic Anomalies of Finite Symmetries on Lattices
AU - Ameya Chavda
AU - Ryohei Kobayashi
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2608.12455
UR - https://arxiv.org/abs/2608.12455
AB - 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 ℤ_2^F→ G_f→ 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×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,ℤ_2) anomaly layer in continuum QFT. In particular, for G_b=ℤ_2, exact lattice symmetries realize only the even ℤ_4 subgroup of the continuum 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×ℤ_2^F, it also forbids a symmetric invertible state. We also construct a non-onsite ℤ_4^F lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with c_-∈ℤ, therefore forbids any invertible states compatible with onsite ℤ_4^F symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.
ER -