General Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb–Schultz–Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors (Ogata and Tasaki, Commun. Math. Phys. 372 951–962, (2019) https://doi.org/10.1007/s00220-019-03343-5 ). We then prove a Lieb–Schultz–Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.