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

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.

To appear in Physical Review E

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• Nonperturbative Quantum Gravity
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• Finite-size scaling above the upper critical dimension in Ising models with long-range interactions
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Cited by in corpus (6)

• Self-Assembly of Geometric Space from Random Graphs
• Random Holographic "Large Worlds" with Emergent Dimensions
• Emergent classical geometries on boundaries of randomly connected tensor networks
• Dynamics and the Emergence of Geometry in an Information Mesh
• Topological Network Entanglement as Order Parameter for the Emergence of Geometry
• Mother canonical tensor model