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arXiv:2610.25884

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

Kai-Wei Huang, Chien-Ting Chen, Shang-Wei Li, Fu-Jiun Jiang

Published Oct 19, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2610.25884

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Phase transitions of the antiferromagnetic J1J_1J1​-J2J_2J2​-J3J_3J3​ Ising model on the cubic lattice
Motivated by the conflicting findings in prior studies regarding the nature of the phase transition for the three-dimensional antiferromagnetic J1J_1J1​-J2J_2J2​-J3J_3J3​ Ising model on the cubic lattice with J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0, in this study we conduct a detailed investigation of this model using both Monte Carlo simulations and neural networks (NN). Here J1J_1J1​ and J2J_2J2​ are the intralayer nearest-neighbor and next-nearest-neighbor couplings, respectively, and J3J_3J3​ 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 J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0 is second order without any ambiguity. Hence we bring a resolution to this debate. Apart from J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0, we study the critical phenomenon associated with J2/J1=4/7J_2/J_1 = 4/7J2​/J1​=4/7 and J3=J1=1.0J_3 = J_1 = 1.0J3​=J1​=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 LLL 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.
We study the phase transitions of the two-dimensional antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0. The method of supervised neural networks (NN) is employed for the investigation. While supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations having their spins be arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ are computed accurately by the resulting NN, but also the nature of the investigated phase transitions are determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations is highly efficient. It remains to examine whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground state configurations.
Abstract We study the phase transitions of the 2D antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0. The method of supervised neural networks (NNs) is employed for the investigation. Although supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations which have their spins arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only are the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ computed accurately by the resulting NN, but also the nature of the investigated phase transitions is determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations are highly efficient. It remains to be examined whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground-state configurations.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
The Monte Carlo method was used to study the phase transitions and thermodynamic properties of the three-dimensional antiferromagnetic Ising model on a layered triangular lattice at various values of interlayer exchange interaction. A number of investigations have been carried out for the ratios of the intralayer (J1) and interlayer (J2) exchange interactions within the range of r = J2/J1 = 0.01–1.0. The nature of the phase transitions was studied using the fourth-order Binder cumulant method and histogram data analysis. It has been shown that in the considered range r, the transition from a disordered phase to an ordered phase is a second-order phase transition.
Based on the replica algorithm by the Monte Carlo method, a computer simulation of the three-dimensional antiferromagnetic Heisenberg model is performed, taking into account the interactions of the first and second nearest neighbors. The phase transitions of this model are studied. The investigations were carried out for the ratios of the exchange interactions of the first and second nearest neighbors r=J2/J1r = J_2 / J_1r=J2​/J1​ in the range 0.0≤r≤1.00.0 \leq r \leq 1.00.0≤r≤1.0. The phase diagram of the critical temperature dependence on a value of the next-nearest neighbor interaction is plotted.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.