Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

TL;DR

The work develops a concrete lattice realization of 't Hooft anomalies in 1+1d, incorporating internal symmetries and Lieb–Schultz–Mattis (LSM) type mixed anomalies with lattice translation. It defines an anomaly cocycle F: G×G×G→U(1) whose cohomology data split into $[\ω\]∈H^3(G,U(1))$ and $[\α\]∈H^2(G,U(1))$, and shows these components obstruct gauging internal symmetries on the lattice unless they vanish in the appropriate cohomology. A systematic gauging procedure using topological defects is presented, with a precise correspondence between locality-preserving symmetry operators and topological defects; this mechanism clarifies how anomalies constrain gauging and propagate to mixed symmetry constraints. The framework is illustrated through bosonic and fermionic examples, including the Heisenberg chain, gauged XYZ chain with non-invertible translation, and the modified Villain model for lattice chiral gauge theories, and it extends to fermionic theories and potential non-invertible spacetime symmetries. Overall, the paper connects lattice anomalies to their continuum counterparts via anomaly matching and provides a versatile toolkit for constructing anomaly-free lattice gauge theories and understanding gauging obstructions.

Abstract

We study 't Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the "anomaly cocycle" using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group $H^3(G,U(1)) \times H^2(G,U(1))$. The first factor captures the anomaly in the internal symmetry group $G$, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving $G$ and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.