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 unique gapped ground state (or, more generally, pure split state), quantum spin chains with on-site symmetry. first prove theorem translation invariant unifies and extends two proved by authors (Ogata Tasaki, Commun. Math. Phys. 372 951–962, (2019) https://doi.org/10.1007/s00220-019-03343-5 ). then are under reflection about origin not necessarily invariant.