Properties of quantum graphity at low temperature

Francesco Caravelli and Fotini Markopoulou
Phys. Rev. D 84, 024002 – Published 5 July 2011

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.

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  • Received 7 April 2011

DOI:https://doi.org/10.1103/PhysRevD.84.024002

© 2011 American Physical Society

Authors & Affiliations

Francesco Caravelli1,2,* and Fotini Markopoulou1,2,3,†

  • 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5 Canada
  • 2University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • 3Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, Golm, D-14476 Golm, Germany

  • *fcaravelli@perimeterinstitute.ca
  • fmarkopoulou@perimeterinstitute.ca

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Vol. 84, Iss. 2 — 15 July 2011

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