Anomalies of Global Symmetries on the Lattice

of global symmetries play a fundamental role in quantum many-body systems and quantum field theory (QFT). In this paper, we make a systematic analysis of —the analog of ’t Hooft anomalies in lattice systems—for which we give a precise definition. Crucially, a lattice anomaly is not a feature of a specific Hamiltonian but rather is a topological invariant of the symmetry action. The controlled setting of lattice systems allows for a systematic and rigorous treatment of lattice anomalies, shorn of the technical challenges of QFT. We find that lattice anomalies reproduce the expected properties of QFT anomalies in many ways but also have crucial differences. In particular, lattice anomalies and QFT anomalies are not, contrary to a common expectation, in one-to-one correspondence, and there can be nontrivial anomalies on the lattice that are : They admit symmetric trivial gapped ground states and map to trivial QFT anomalies at low energies. Nevertheless, we show that lattice anomalies (including IR-trivial ones) have a number of interesting consequences in their own right, including connections to commuting projector models, phases of many-body-localized systems, and quantum cellular automata. We make substantial progress on the classification of lattice anomalies and develop several theoretical tools to characterize their consequences on symmetric Hamiltonians. Our work places symmetries of quantum many-body lattice systems into a unified theoretical framework and may also suggest new perspectives on symmetries in QFT.