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

We investigate the self-organization of point-particles with short-range interactions modeled via simple 1D and 2D Hubbard-like models. We show how various properties emerge such as, boson-like ordering leading to topological structures in real space, via the clustering of the particles at discrete energy levels, which can be analyzed using a network/graph mathematical language. We calculate analytically the number of clusters, the correlations between them and topological numbers like the Euler characteristic, deriving different organizational schemes and entanglement entropy scaling laws. All calculations are performed for an arbitrary number of particles N and energy of the system E. Our results demonstrate how orders with diverse topological/geometric and entanglement features, emerge in strongly interacting many-body systems, that follow minimal microscopic interaction rules.