Complex Quantum Network Geometries: Evolution and Phase Transitions

TL;DR

This paper introduces complex quantum network geometries based on evolving simplicial complexes, revealing their properties, statistical behaviors, and phase transitions, with implications for quantum space-time and network science.

Contribution

It defines quantum geometric networks with non-equilibrium dynamics, characterizes their properties, and demonstrates their phase transitions, linking quantum states to network topology.

Findings

Quantum geometric networks exhibit small-world and scale-free properties.

Networks can obey Fermi-Dirac or Bose-Einstein statistics.

Structural phase transitions drastically change network geometry.

Abstract

Networks are topological and geometric structures used to describe systems as different as the Internet, the brain or the quantum structure of space-time. Here we define complex quantum network geometries, describing the underlying structure of growing simplicial 2-complexes, i.e. simplicial complexes formed by triangles. These networks are geometric networks with energies of the links that grow according to a non-equilibrium dynamics. The evolution in time of the geometric networks is a classical evolution describing a given path of a path integral defining the evolution of quantum network states. The quantum network states are characterized by quantum occupation numbers that can be mapped respectively to the nodes, links, and triangles incident to each link of the network. We call the geometric networks describing the evolution of quantum network states the quantum geometric networks. The quantum geometric networks have many properties common to complex networks including small-world property, high clustering coefficient, high modularity, scale-free degree distribution.Moreover they can be distinguished between the Fermi-Dirac Network and the Bose-Einstein Network obeying respectively the Fermi-Dirac and Bose-Einstein statistics. We show that these networks can undergo structural phase transitions where the geometrical properties of the networks change drastically. Finally we comment on the relation between Quantum Complex Network Geometries, spin networks and triangulations.