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.Received 22 December 2021Revised 2 May 2022Accepted 26 May 2022DOI:https://doi.org/10.1103/PhysRevX.12.031007Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCharge density wavesCrystal symmetryExotic phases of matterQuantum entanglementQuantum gatesTopological phases of matterCondensed Matter, Materials & Applied PhysicsQuantum Information