Properties of Quantum Graphity at Low Temperature

TL;DR

This paper maps Quantum Graphity models to an Ising Hamiltonian to analyze their low-temperature properties, revealing how average valence acts as an order parameter and deriving its temperature dependence.

Contribution

It introduces a novel mapping of Quantum Graphity models to Ising Hamiltonians, enabling analytical study of low-temperature behavior and valence distribution.

Findings

Average valence serves as an order parameter.

Derived the temperature dependence of valence.

Provided low-temperature expansion coefficients.

Abstract

We present a mapping of dynamical graphs and, in particular, the graphs used in the Quantum Graphity models for emergent geometry, into an Ising hamiltonian on the line graph of a complete graph with a fixed number of vertices. We use this method to study the properties of Quantum Graphity models at low temperature in the limit in which the valence coupling constant of the model is much greater than the coupling constants of the loop terms. Using mean field theory we find that an order parameter for the model is the average valence of the graph. We calculate the equilibrium distribution for the valence as an implicit function of the temperature. In the approximation in which the temperature is low, we find the first two Taylor coefficients of the valence in the temperature expansion. A discussion of the susceptibility function and a generalization of the model are given in the end.