2D additive small-world network with distance-dependent interactions

In this work, we have employed Monte Carlo calculations to study the Ising model on a 2D additive small-world network with long-range interactions depending on the geometric distance between interacting sites. The network is initially defined by a regular square lattice and with probability p each site is tested for the possibility of creating a long-range interaction with any other site that has not yet received one. Here, we used the specific case where p=1, meaning that every site in the network has one long-range interaction in addition to the short-range interactions of the regular lattice. These long-range interactions depend on a power-law form, J₈₉=r₈₉^-, with the geometric distance r₈₉ between connected sites i and j. In current two-dimensional model, we found that mean-field critical behavior is observed only at =0. As increases, the network size influences the phase transition point of the system, i. e. , indicating a crossover behavior. However, given the two-dimensional system, we found the critical behavior of the short-range interaction at 2. Thus, the limitation in the number of long-range interactions compared to the globally coupled model, as well as the form of the decay of these interactions, prevented us from finding a regime with finite phase transition points and continuously varying critical exponents in 0<<2.