Non-equilibrium Ising Model on a 2D 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 a $L\times L$ 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 $\textrm{m}_{\textrm{L}}^{\textrm{F}}$ and staggered $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ magnetizations per spin, the susceptibility $\chi_{\textrm{L}}$, and the reduced fourth-order Binder cumulant $\textrm{U}_{\textrm{L}}$. 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.