Enhancement of Tc in the superconductor–insulator phase transition on scale-free networks

A road map for understanding the relation between the onset of the superconducting state with a particular optimum heterogeneity in granular superconductors is provided by studying a random transverse Ising model on complex networks with a scale-free degree distribution regularized by an exponential cutoff p(k) ∝ k−γexp[ − k/ξ]. In this paper we characterize in detail the phase diagram of this model, both on annealed and on quenched networks. To uncover the phase diagram of the model we use the tools of heterogeneous mean-field calculations for the annealed networks and the most advanced techniques of quantum cavity methods for the quenched networks. The phase diagram of the dynamical process depends on the temperature T, on the coupling constant J and on the value of the branching ratio 〈k(k − 1)〉/〈k〉 where k is the degree of the nodes in the network. For a fixed value of the coupling, the critical temperature increases linearly with 〈k(k − 1)〉/〈k〉, which diverges with increasing cutoff value ξ for values of the γ exponent γ ≤ 3. This result suggests that the fractal disorder of the superconducting material can be responsible for an enhancement of the superconducting critical temperature. For low temperature and low couplings T ≪ 1 and J ≪ 1, however, we observe different behaviors for annealed and quenched networks. In the annealed networks there is no phase transition at zero temperature, while in the quenched network a Griffith phase dominated by extremely rare events and a phase transition at zero temperature are observed. The Griffiths critical region, however, decreases in size with increasing value of the cutoff ξ of the degree distribution for values of the γ exponents γ ≤ 3. (paper)