Abstract
When two non-relativistic particles interact resonantly in three dimensions, an infinite tower of three-body bound states emerges, exhibiting a discrete scale invariance. This universal phenomenon, known as the Efimov effect, has garnered extensive attention across various fields. However, it remains an open question how analogous universal few-body physics can emerge in low-dimensional quantum platforms. In this work, we demonstrate that the Efimov effect also manifests in long-range quantum spin chains. The long-range coupling modifies the low-energy dispersion of magnons, enabling the emergence of continuous scale invariance for two-magnon states at resonance. This invariance is subsequently broken to discrete scale invariance for the three-magnon problem, leading to the celebrated Efimov effect. We further discuss generalizations to arbitrary spatial dimensions, where the traditional Efimov effect serves as a special case. Our results reveal universal physics in dilute quantum gases of magnons that can be experimentally tested in trapped-ion systems.
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Data availability
The Supplementary Data for Fig. 3 generated in this study are provided in https://github.com/atom-sun/LongrangeEfimov/data.
Code availability
The custom code used to generate the analysis and figures in this manuscript is available on GitHub at https://github.com/atom-sun/LongrangeEfimov.
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Acknowledgements
We thank Zhenhua Yu for helpful discussions. This project is supported by the Shanghai Municipal Science and Technology Major Project Grant No. 24DP2600100 (NS and LF), Co-research Program under Grant No. 25LZ2601000 (NS and LF), the Quantum Science and Technology–National Science and Technology Major Project under Grant No. 2023ZD0300900 (LF), 2024ZD0300101 (PZ), 2025ZD0300100 (NS and LF) and 2025ZD0300101 (NS and LF), the Shanghai Rising-Star Program under grant number 24QA2700300 (PZ), and the NSFC under grant 12374477 (PZ).
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N.S. conducted the analytical calculation and numerical verification. L.F. and P.Z. proposed the study and supervised the project. All authors contributed to discussing the results and writing the manuscript.
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Sun, N., Feng, L. & Zhang, P. Efimov effect in long-range quantum spin chains. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02580-0
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DOI: https://doi.org/10.1038/s42005-026-02580-0
