Non-equilibrium Ising Model on a 2D Additive Small-World Network

TL;DR

This study investigates the non-equilibrium Ising model on a 2D additive small-world network, revealing phase transitions and universality class changes due to network topology and competing dynamics.

Contribution

It introduces a novel non-equilibrium Ising model on an additive small-world network with competing dynamics, analyzing phase diagrams and critical behavior via Monte Carlo simulations.

Findings

Identified phase transition lines between ferromagnetic, paramagnetic, and antiferromagnetic phases.

Showed the phase diagram topology changes with increasing probability p.

Discovered a shift in universality class from regular lattice to small-world network.

Abstract

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.