In this talk, I will show how crystalline momentum holds a deep connection to quantum entanglement and quantum anomalies. First I will demonstrate that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero crystalline momentum . The statement can also be generalized to fermion systems, and all dimensions. I will present several non-trivial consequences that follow from our theorem:
• Several different types of Lieb-Schultz-Mattis-Oshikawa-Hastings theorems, including a previously unknown version involving only a discrete symmetry;
• 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;
• Quantum anomalies associated to momentum are described by a Chern-Simons term that emerges in physical systems such as the time-reversal invariant Weyl semimetals .
I conclude by presenting some potential generalisations of this statement, and work that remains to be done especially regarding the possible roles of momentum in emergent symmetries.
 Lei Gioia, Chong Wang, Phys. Rev. X 12, 031007 (2022)
 Lei Gioia, Chong Wang, A. A. Burkov, Phys. Rev. R 3, 043067 (2021)