Mass generation in graphs

TL;DR

This paper introduces a Higgs-inspired mechanism in graphs that produces massive excitations, whose properties depend on the graph's size and density, illustrating emergent matter-like structures in discrete models.

Contribution

It presents a novel graph-based mass generation mechanism inspired by quantum field theory, linking graph connectivity to emergent massive particles.

Findings

Massive excitations localize on high-density regions of the graph.

The mass gap and localization depend on the graph's size and density.

Emergent matter-like structures can arise from simple connectivity properties.

Abstract

We demonstrate a mechanism for the production of massive excitations in graphs. We treat the number of neighbors at each vertex in the graph (degree) as a scalar field. Then we introduce a mechanism inspired by the Higgs mechanism in quantum field theory(QFT), that couples the degree field to a vector-like field, introduced via the graph edges, represented mathematically by the incident matrices of the graph. The coupling between the two fields produces a massless ground state and massive excitations, separated by a mass gap. The excitations can be treated as emergent massive particles, propagating inside the graph. We study how the size of the graph and its density, represented by the ratio of edges over vertices, affects the mass gap and the localization properties of the massive excitations. We show that the most massive excitations, corresponding to the heaviest emergent particles, localize on regions of the graph with high density, consisting of vertices with a large degree. On the other hand, the least massive excitations, corresponding to the lightest emergent particles localize on a few vertices but with a smaller degree. Excitations with intermediate masses are less localized, spreading on more vertices instead. Our study shows that emergence of matter-like structures with various mass properties, is possible in discrete physical models, relying only on a few fundamental properties like the connectivity of the models.