Critical Space-Time Networks and Geometric Phase Transitions from Frustrated Edge Antiferromagnetism

TL;DR

This paper analyzes a simplified bipartite graph model of space-time that exhibits a geometric phase transition linked to antiferromagnetic edge interactions, revealing a critical point with long-range correlations and continuum geometry.

Contribution

It provides an exact solution of a toy model connecting geometric phase transitions to antiferromagnetic edge behavior in a bipartite graph setting.

Findings

The geometric phase transition aligns with the antiferromagnetic transition.

A critical point with long-range correlations enables continuum geometry.

The model generalizes Kazakov's 2D lattice results to higher dimensions.

Abstract

Recently I proposed a simple dynamical network model for discrete space-time which self-organizes as a graph with Hausdorff dimension d_H=4. The model has a geometric quantum phase transition with disorder parameter (d_H-d_s) where d_s is the spectral dimension of the dynamical graph. Self-organization in this network model is based on a competition between a ferromagnetic Ising model for vertices and an antiferromagnetic Ising model for edges. In this paper I solve a toy version of this model defined on a bipartite graph in the mean field approximation. I show that the geometric phase transition corresponds exactly to the antiferromagnetic transition for edges, the dimensional disorder parameter of the former being mapped to the staggered magnetization order parameter of the latter. The model has a critical point with long-range correlations between edges, where a continuum random geometry can be defined, exactly as in Kazakov's famed 2D random lattice Ising model but now in any number of dimensions.