Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2010-08-07
Phys.Rev.D84:024002,2011
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
21 pages, 5 figures; a plot added, corrections added; Accepted for publication in PRD
Scientific paper
10.1103/PhysRevD.84.024002
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.
Caravelli Francesco
Markopoulou Fotini
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