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We investigate the self-organization of point-particles with short-rangeinteractions modeled via simple 1D and 2D Hubbard-like models. We show howvarious properties emerge such as, boson-like ordering leading to topologicalstructures in real space, via the clustering of the particles at discreteenergy levels, which can be analyzed using a network/graph mathematicallanguage. We calculate analytically the number of clusters, the correlationsbetween them and topological numbers like the Euler characteristic, derivingdifferent organizational schemes and entanglement entropy scaling laws. Allcalculations are performed for an arbitrary number of particles N and energy ofthe system E. Our results demonstrate how orders with diversetopological/geometric and entanglement features, emerge in strongly interactingmany-body systems, that follow minimal microscopic interaction rules.

Self-organization in many-body systems with short-range interactions: clustering, correlations and topology