A result by Gioia and Wang [Phys. Rev. X 12, 031007 (2022)] showed that translationally symmetric states having nonzero momentum are necessarily long-range entangled (LRE). Here, we consider the following question: Can a notion of momentum for nontranslation-symmetric states directly encode the nature of their entanglement, as it does for translation-symmetric states? We show the answer is affirmative for one-dimensional (1D) systems, while higher-dimensional extensions and topologically ordered systems require further work. While the Gioia and Wang result applies to states connected via finite-depth quantum circuits to a translation-symmetric state, it is often impractical to find such a circuit to determine the nature of the entanglement of states that break translation symmetry. Here, instead of translation eigenstates, we focus on the many-body momentum distribution and the expectation value of the translation operator in many-body states of systems having broken translation symmetry. We show that in the continuum limit the magnitude of the expectation value of the translation operator $|\langle T\rangle |$ necessarily goes to 1 for delocalized states, a proxy for LRE states in 1D systems. This result can be seen as a momentum-space version of Resta's formula for the localization length. We investigate how accurate our results are in different lattice models with and without well-defined continuum limits. To that end, we introduce two models: a deterministic version of the random dimer model, illustrating the role of the thermodynamic and continuum limits for our result at a lattice level, and a simplified version of the Aubry-Andre model, with commensurate hopping for both momentum and position space. Finally, we use the random dimer model as a test case for the accuracy of $|\langle T\rangle |$ as a localization (and thus entanglement) probe for 1D periodic lattice models without a well-defined continuum limit.

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• Entanglement measures
• 1-dimensional systems
• Disordered systems
• Lattice models in condensed matter
• Symmetries in condensed matter