Phase transitions of the antiferromagnetic $J_1$-$J_2$-$J_3$ Ising model on the cubic lattice

Abstract

Motivated by the conflicting findings in prior studies regarding the nature of the phase transition for the three-dimensional antiferromagnetic $J_1$-$J_2$-$J_3$ Ising model on the cubic lattice with $J_1 = J_2 = J_3 = 1.0$, in this study we conduct a detailed investigation of this model using both Monte Carlo simulations and neural networks (NN). Here $J_1$ and $J_2$ are the intralayer nearest-neighbor and next-nearest-neighbor couplings, respectively, and $J_3$ is the nearest-neighbor interlayer interaction. By investigating the data obtained from much larger lattices than those used in previous studies, we find that the phase transition for $J_1 = J_2 = J_3 = 1.0$ is second order without any ambiguity. Hence we bring a resolution to this debate. Apart from $J_1 = J_2 = J_3 = 1.0$, we study the critical phenomenon associated with $J_2/J_1 = 4/7$ and $J_3 = J_1 = 1.0$ as well. In addition, by employing a dedicated training and testing strategy, our NN computation not only is much more efficient than those utilizing the standard procedures commonly used in the literature, but also requires only a per-mille fraction of the storage capacity needed by a conventional NN calculation when a large linear system size $L$ is considered. The method employed here provides an extremely resource-efficient route for applying NN techniques to study the critical behaviors of higher-dimensional systems.