Non-zero momentum requires long-range entanglement

TL;DR

The paper proves that quantum states with non-zero lattice momentum must exhibit long-range entanglement, leading to new insights into topological order, symmetry-breaking, and classification of crystalline phases.

Contribution

It establishes a fundamental link between non-zero momentum and long-range entanglement, deriving new theorems and implications for topological order and symmetry in quantum systems.

Findings

States with non-zero momentum are necessarily long-range entangled.

New versions of Lieb-Schultz-Mattis-Oshikawa-Hastings theorems are derived.

Gapped topological orders with non-zero momentum must weakly break translation symmetry.

Abstract

We show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero lattice momentum, i.e. if it is an eigenstate of the translation symmetry with eigenvalue $e^{iP}\neq1$. Equivalently, any state that can be connected with a non-zero 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 non-trivial consequences follow immediately from our theorem: (1) several different types of Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems, including a previously unknown version involving only a discrete $\mathbb{Z}_n$ symmetry, can be derived in a simple manner from our result; (2) 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; (3) our result provides further evidence of the "smoothness" assumption widely used in the classification of crystalline symmetry-protected topological (cSPT) phases.