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

TL;DR

This paper explores how simple short-range interactions in many-body systems lead to complex self-organized structures, topological features, and entanglement properties, analyzed through analytical models in 1D and 2D.

Contribution

It introduces analytical methods to study clustering, correlations, and topology in many-body systems with short-range interactions, revealing emergent order and entanglement scaling laws.

Findings

Particles form topological structures via clustering at discrete energy levels

Analytical formulas for number of clusters and topological invariants derived

Entanglement entropy exhibits specific scaling laws in these systems

Abstract

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.