Persistent self-organized states in non-equilibrium magnetic models

In this work, we employed Monte Carlo simulations to study the Ising, $XY$, and Heisenberg models on a simple cubic lattice, where the system models evolve toward the steady state under the influence of competition between one- and two-spin flip dynamics. With probability $q$, the system is in contact with a thermal reservoir at temperature $T$ and evolves toward the lower energy state through one-spin flip dynamics. On the other hand, with probability $1-q$, the system is subjected to an external energy flux that drives it toward the higher energy state through two-spin flip dynamics. As a result, we constructed the phase diagram of $T$ as a function of $q$. In this diagram, we identified the antiferromagnetic ($AF$) ordered phase, the ferromagnetic ($F$) ordered phase, and the disordered paramagnetic ($P$) phase for all the models studied. Through these phases, we observed self-organization phenomena in the systems. For low values of $q$, the system is in the $AF$ phase, and as $q$ increases the system continuously transitions to the $P$ phase. Now, for high values of $q$, the system through continuous phase transitions again reaches an ordered phase, the $F$ phase, at low values of $T$. Additionally, we also calculated the critical exponents of the system, showing that these are not affected by the non-equilibrium regime of the system.