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.

中文翻译：

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具有短程相互作用的多体系统中的自组织：聚类、相关性和拓扑

我们研究了通过简单的一维和二维类似哈伯德模型建模的具有短程相互作用的点粒子的自组织。我们展示了各种属性是如何出现的，例如，通过离散能级粒子的聚类，在真实空间中导致拓扑结构的类玻色子排序，可以使用网络/图形数学语言进行分析。我们分析计算集群的数量，它们之间的相关性和拓扑数（如欧拉特性），推导出不同的组织方案和纠缠熵缩放定律。所有计算都是针对任意数量的粒子 N 和系统 E 的能量进行的。我们的结果证明了具有不同拓扑/几何和纠缠特征的阶是如何出现在强相互作用的多体系统中的，