Critical behavior of the XY model in complex topologies

The critical behavior of the O(2) model on dilute Lévy graphs built on a two-dimensional square lattice is analyzed. Different qualitative cases are probed, varying the exponent ρ governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes, are investigated by means of high-performance parallel Monte Carlo numerical simulations running on GPUs. The relationship between the long-range ρ exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the Lévy graph for the XY model in the mean-field regime. © 2013 American Physical Society.