Critical behavior of the XY model on uncorrelated and correlated random networks

Abstract

We numerically study the critical behavior of the XY model on the Erdős–Rényi random graph and a growing random network model, representing the uncorrelated and the correlated random networks, respectively. We also checked the dependence of the critical behavior on the choice of order parameters: the ordinary unweighted and the degree-weighted magnetization. On the Erdős–Rényi random network, the critical behavior of the XY model is found to be of the second order with the estimated exponents consistent with the standard mean-field theory for both order parameters. On the growing random network, on the contrary, we found that the critical behavior is not of the standard mean-field type. Rather, it exhibits behavior reminiscent of that in the infinite-order phase transition for both order parameters, such as the lack of discontinuity in specific heat and the non-divergent susceptibility at the critical point, as observed in the percolation and the Potts models on some growing network models.