Emergence of Long Range Order in the XY Model on Diluted Small World Networks

Abstract

We study the XY model on diluted Small World networks, i.e Small World networks whose number of links scales with the system size N _links ∼N ^γ, 1<γ<2. Starting from the regular lattice topology, we first concentrate on the behaviour varying the dilution parameter γ: for low values, the system does not exhibit a phase transition; while for γ approaching 2 a second order transition of the magnetisation arises since the system is near the HMF regime. Hence γ _c =1.5 appears to be a critical value: an energy range is observed in which the magnetisation shows important fluctuations and does not reach the equilibrium state. We then take in account the model on a Small World network: for the latter, we have chosen the Watts-Strogatz model, whose topology is parametrized by the rewiring probability p, 0<p<1. We performed microcanonical simulations of the dynamics and we highlight the presence of a second order phase transition appearing even for very low p and γ, when the topology is still near the regular lattice one. Moreover we observe a dependence of the critical energy ϵ _c on the rewiring probability p.