We show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has nonzero lattice momentum, i.e., if it is an eigenstate of the translation symmetry with eigenvalue $e^{iP}\ne 1$. Equivalently, any state that can be connected with a nonzero momentum state through a finite-depth local unitary transformation must also be long-range entangled. The statement can also be generalized to fermion systems. Some nontrivial consequences follow immediately from our theorem: (i) Several different types of Lieb-Schultz-Mattis-Oshikawa-Hastings theorems, including a previously unknown version involving only a discrete ${\mathbb{Z}}_n$ symmetry, can be derived in a simple manner from our result; (ii) a gapped topological order (in space dimension $d>1$) must weakly break translation symmetry if one of its ground states on torus has nontrivial momentum—this generalizes the familiar physics of Tao-Thouless; (iii) our result provides further evidence of the “smoothness” assumption widely used in the classification of crystalline symmetry-protected topological phases.

中文翻译：

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非零动量需要长距离纠缠

我们证明，如果晶格自旋（玻色子）系统中的量子态具有非零晶格动量，则它必须是长程纠缠的，即，如果它是具有特征值的平移对称的特征态$e^{\mathrm{一世}磷}\ne 1$. 等效地，任何可以通过有限深度的局部酉变换与非零动量状态连接的状态也必须是长程纠缠的。该陈述也可以推广到费米子系统。我们的定理会立即得出一些重要的结果： (i) 几种不同类型的 Lieb-Schultz-Mattis-Oshikawa-Hastings 定理，包括一个以前未知的版本，只涉及一个离散的${\mathbb{Z}}_n$对称性，可以从我们的结果中以简单的方式推导出来；(ii) 有间隙的拓扑顺序（在空间维度上$d>1$)如果其在环面上的一个基态具有非平凡动量，则必须弱破坏平移对称性——这概括了熟悉的 Tao-Thouless 物理学；(iii) 我们的结果进一步证明了广泛用于晶体对称保护拓扑相分类的“平滑”假设。