Superconductor-insulator transition on annealed complex networks

Cuprates show multiphase and multiscale complexity that has hindered physicists search for the mechanism of high T-c for many years. Recently the interest has been addressed to a possible optimum inhomogeneity of dopants, defects, and interstitials, and the structural scale invariance of dopants detected by scanning micro-x-ray diffraction has been reported to promote the critical temperature. In order to shed light on critical phenomena on granular materials, here we propose a stylized model capturing the essential characteristics of the superconducting-insulator transition of a highly dynamical, heterogeneous granular material: the random transverse Ising model (RTIM) on annealed complex network. We show that when the networks encode for high heterogeneity of the expected degrees described by a power-law distribution, the critical temperature for the onset of the superconducting phase diverges to infinity as the power-law exponent gamma of the expected degree distribution is less than 3, i.e., gamma < 3. Moreover we investigate the case in which the critical state of the electronic background is triggered by an external parameter g that determines an exponential cutoff in the power-law expected degree distribution characterized by an exponent gamma. We find that for g = g(c) the critical temperature for the superconducting-insulator transition has a maximum if gamma > 3 and diverges if gamma < 3.