Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem

For any locality-preserving action of a group $G$ on a quantum spin chain onecan define an anomaly index taking values in the group cohomology of $G$. Theanomaly index is a kinematic quantity, it does not depend on the Hamiltonian.We prove that a nonzero anomaly index prohibits any $G$-invariant Hamiltonianfrom having $G$-invariant gapped ground states. Lieb-Schultz-Mattis-typetheorems are a special case of this result when $G$ involves translations. Inthe case when the symmetry group $G$ is a Lie group, we define an anomaly indexwhich takes values in the differentiable group cohomology as defined by J.-L.Brylinski and prove a similar result.