Properties of quantum graphity at low temperature

Abstract

We present a mapping of dynamical graphs and, in particular, the graphs used in the Quantum Graphity models for emergent geometry, to 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.

Energetics of the quantum graphity universe

“…Edge creation or deletion is therefore a non-energy conserving process. As mentioned above, this implies some form for thermal reservoir is necessary to force evolution to the ground state [3], which may be problematic as the graph represents the entire Universe. This problem is addressed by associating the loss of energy of the graph with the creation of matter (and vice versa, the annihilation of matter can raise the energy of the graph) [2].…”

“…It is therefore necessary to posit mechanisms by which the graph can relax to some global minimum, which is presumed to have the properties of geometry that we experience. Mechanisms to allow such relaxation include the formation of matter on the graph [2], and equilibration with some external heat bath [3].…”

Geometrogenesis under quantum graphity: Problems with the ripening universe

“…This graph has no geometrical interpretation, making it a candidate for pre-geometry. Furthermore, thermodynamic arguments support the notion that higher temperature graphs have more edges, making the graph with the most edges possible a natural maximum temperature limit [13]. However, the empty graph, in which there exist vertices but no edges, so that every point in space is completely isolated, is also a candidate high-temperature pre-geometric graph [14].…”

“…When recast on a line graph, the model can be reduced to an analogue of an Ising model, making it amenable to a mean-field treatment. In this approach, using the average degree of the graph as an order parameter, it was found that as the number of vertices goes to infinity, the critical temperature of the geometrogenic phase transition goes to zero [13]. Another approach has been to estimate the partition function of the model by considering which types of graphs will contribute most [15].…”

Domain structures in quantum graphity

“…Previous works have studied how locality emerges in the model [16], the role matter plays in the emergence of extended geometries [17], the entanglement of matter with spatial degrees of freedom [18], Ising mappings to study low temperature properties [19], and the entrapment of matter in regions of high connectivity [20]. See Ref.…”