Nonequilibrium Ising model on a two-dimensional additive small-world network

In this work, we have studied the Ising model with one- and two-spin flip competing dynamics on a two-dimensional additive small-world network (A-SWN). The system model consists of an L×L square lattice where each site of the lattice is occupied by a spin variable that interacts with the nearest-neighbor spins and it has a certain probability p of being additionally connected at random to one of its farther neighbors. The dynamics present in the system can be defined by the probability q of being in contact with a heat bath at a given temperature T and, at the same time, with a probability of (1-q) the system is subjected to an external flux of energy into the system. The contact with the heat bath is simulated by one-spin flip according to the Metropolis prescription, while the input of energy is mimicked by the two-spin flip process, involving a simultaneous flipping of a pair of neighboring spins. We have employed Monte Carlo simulations to obtain the thermodynamic quantities of the system, such as the total m_{L}^{F} and staggered m_{L}^{AF} magnetizations per spin, the susceptibility χ_{L}, and the reduced fourth-order Binder cumulant U_{L}. We have built the phase diagram for the stationary states of the model in the plane T versus q, showing the existence of two continuous transition lines for each value of p: one line between the ferromagnetic F and paramagnetic P phases and the other line between the P and antiferromagnetic AF phases. Therefore, we have shown that the phase diagram topology changes when p increases. Using the finite-size scaling analysis, we also obtained the critical exponents for the system, where, varying the parameter p, we have observed a different universality class from the Ising model in the regular square lattice to the A-SWN.