https://doi.org/10.1140/epja/s10050-025-01736-w
Review
Clustering as a window on the hierarchical structure of quantum systems
1
Department of Physics, Institute of Science Tokyo, 2-12-1 O-Okayama, Meguro, 152-8550, Tokyo, Japan
2
Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, 1-3-1 Kagamiyama, 739-8526, Higashi-Hiroshima, Hiroshima, Japan
3
Research Center for Accelerator and Radioisotope Science (RARiS), Tohoku University, 1-2-1 Mikamine, Taihaku-ku, 982-0826, Sendai, Miyagi, Japan
4
Department of Physics, Tohoku University, Aramaki, Aoba-ku, 980-8578, Sendai, Miyagi, Japan
5
Department of Physics, Graduate School of Science, Kyoto University, 606-8502, Kyoto, Japan
6
Faculty of Science, Osaka Metropolitan University, 3-3-138 Sugimoto-cho, Sumiyoshi-ku, 558-8585, Osaka, Japan
7
Research Center for Nuclear Physics (RCNP), Osaka University, 567-0048, Ibaraki, Osaka, Japan
a
nakamura@phys.sci.isct.ac.jp
Received:
8
October
2024
Accepted:
27
October
2025
Published online:
1
December
2025
Why do quantum particles form a hierarchical structure: quarks, hadrons, nuclei, atoms, and molecules? This is a fundamental question, and its answer is still elusive. Each hierarchical layer is characterized by the constituent particles, which are composite particles except for the quark hierarchy. Such a building block is regarded as a cluster and plays a role in forming a hierarchy. In the boundary of the neighboring hierarchies, we may find intermediate hierarchies, called semi-hierarchies, where a range of characteristic clusters, such as hadronic molecules, exotic hadrons, neutron halos, 
clusters, and Feshbach molecules, appear. Such a cluster structure has some common features throughout the hierarchical layers with different scales. We discuss the role of clusters and their formation in semi-hierarchies.
© The Author(s) 2025
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