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 one can define an anomaly index taking values in the group cohomology of G. The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any G-invariant Hamiltonian from having G-invariant gapped ground states. Lieb–Schultz–Mattis-type theorems are a special case of this result when G involves translations. In the case when the symmetry group G is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.