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.

我们开发了一种通用算子代数方法，该方法侧重于对称群的投影表示，以证明 Lieb-Schultz-Mattis 型定理，即排除唯一有隙基态（或更一般地说，纯分裂状态），用于具有现场对称性的量子自旋链。我们首先证明了平移不变自旋链的定理，该定理统一和扩展了由两位作者证明的两个定理（Ogata 和 Tasaki，Commun. Math. Phys. 372 951–962, (2019) https://doi.org/10.1007 /s00220-019-03343-5)。然后，我们证明了自旋链的 Lieb-Schultz-Mattis 类型定理，该定理在关于原点的反射下是不变的，并且不一定是平移不变的。