Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D

[2601.15345] Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Condensed Matter > Disordered Systems and Neural Networks arXiv:2601.15345 (cond-mat) [Submitted on 20 Jan 2026 ( v1 ), last revised 20 Feb 2026 (this version, v2)] Title: Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D Authors: Amanda Gatto Lamas , Taylor L. Hughes View a PDF of the paper titled Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D, by Amanda Gatto Lamas and 1 other authors View PDF HTML (experimental) Abstract: 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 question: can a notion of momentum for non-translation symmetric states directly encode the nature of their entanglement, as it does for translation symmetric states? We show the answer is affirmative for 1D systems, while higher dimensional extensions and topologically ordered systems require further work. While Gioia and Wang's 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 $|<T>|$ 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 $|<T>|$ as a localization (and thus entanglement) probe for 1D periodic lattice models without a well-defined continuum limit. Comments: 24+8 pages, 15+5 figures. Added footnote comparing the "self-dual" model to the Harper Hamiltonian Subjects: Disordered Systems and Neural Networks (cond-mat.dis-nn) ; Quantum Physics (quant-ph) Cite as: arXiv:2601.15345 [cond-mat.dis-nn] (or arXiv:2601.15345v2 [cond-mat.dis-nn] for this version) https://doi.org/10.48550/arXiv.2601.15345 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Phys. Rev. B 114, 024205 (2026) Related DOI : https://doi.org/10.1103/nvpn-7f36 Focus to learn more DOI(s) linking to related resources Submission history From: Amanda Gatto Lamas [ view email ] [v1] Tue, 20 Jan 2026 19:01:06 UTC (3,837 KB) [v2] Fri, 20 Feb 2026 21:13:04 UTC (3,837 KB) Full-text links: Access Paper: View a PDF of the paper titled Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D, by Amanda Gatto Lamas and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.dis-nn < prev | next > new | recent | 2026-01 Change to browse by: cond-mat quant-ph References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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