Nonzero Momentum Requires Long-Range Entanglement

Abstract

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 eiP≠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 Zn 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.