Topological Network Entanglement as Order Parameter for the Emergence of Geometry

TL;DR

This paper demonstrates that in discrete quantum gravity models, the emergence of geometric space correlates with a reduction in topological network entanglement, serving as an order parameter for geometric phase transition.

Contribution

It introduces the concept of topological network entanglement as an order parameter for emergent geometry in quantum gravity models, with analysis of a specific curvature-driven model.

Findings

Emergent geometry correlates with decreased entanglement entropy.

Four-dimensional configurations exhibit the lowest entanglement entropy.

Topological network entanglement characterizes geometric phase transitions.

Abstract

We show that, in discrete models of quantum gravity, emergent geometric space can be viewed as the entanglement pattern in a mixed quantum state of the "universe", characterized by a universal topological network entanglement. As a concrete example we analyze the recently proposed model in which geometry emerges due to the condensation of 4-cycles in random regular bipartite graphs, driven by the combinatorial Ollivier-Ricci curvature. Using this model we show that the emergence of geometric order decreases the entanglement entropy of random configurations. The lowest geometric entanglement entropy is realized in four dimensions.