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Progress of Theoretical and Experimental Physics

A Machine Learning Study of the 2D Antiferromagnetic Ising Model with Nearest and Next-to-Nearest Interactions on the Triangular Lattice

Shang-Wei Li, Yuan-Heng Tseng, Kai-Wei Huang, Fu-Jiun Jiang

Published Jun 30, 2026• 0 citations
Publisher

Abstract

Abstract We study the phase transitions of the 2D antiferromagnetic Ising model with nearest $J_1$ and next-to-nearest $J_2$ interactions on the triangular lattice for $J_2/J_1 = 0.1, 0.5$, 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 $J_2/J_1$ computed accurately by the resulting NN, but also the nature of the investigated phase transitions is determined correctly. Specifically, the phase transitions associated with $J_2/J_1 = 0.1, 0.5$, 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.

DOI
10.1093/ptep/ptag123

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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.
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.
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.
Abstract The critical phenomena of the 2D antiferromagnetic qqq-state Potts model on the square lattice with q=2,3,4q=2,3,4q=2,3,4 are investigated using the techniques of neural networks (NNs). In particular, an unconventional supervised NN which is trained using no information about the physics of the considered systems is employed. In addition, conventional unsupervised autoencoders (AECs) are used in our study as well. Remarkably, whereas the conventional AECs either fail or only work partially to uncover the critical phenomena of the systems associated with q=3q=3q=3 and q=4q=4q=4 investigated here, our unconventional supervised NN correctly identifies the critical behaviors of all three considered antiferromagnetic qqq-state Potts models. The results obtained in this study suggest convincingly that the applicability of our unconventional supervised NN is broader than one anticipates. In particular, when a new system is studied with our NN, it is likely that it is not necessary to conduct any training, and one only needs to examine whether an appropriate reduced representation of the original raw configurations exists, so that the same already trained NN can be employed to explore the related phase transition efficiently.
A Machine Learning study of the two-dimensional antiferromagnetic qqq-state Potts model on the square lattice
The critical phenomena of two-dimensional (2D) antiferromagnetic qqq-state Potts model on the square lattice with q=2,3,4,5q=2,3,4,5q=2,3,4,5 and 6 are investigated using the technique of supervised neural network (NN). Unlike the conventional NN approaches, here we train a multilayer perceptron consisting of only one input layer, one hidden layer, and one output layer with two artificially made stagger-like configurations. Remarkably, despite the fact that the MLP is trained without any input from these considered models, it correctly identifies the critical temperatures of the studied physical systems. Particularly, the MLP outcomes suggest convincingly that the q=3q=3q=3 model is critical only at zero temperature and q=4,5,6q=4,5,6q=4,5,6 models remain disordered at all temperatures. Previously, this MLP has been successfully applied to uncover the nature of the phase transitions of 2D antiferromagnetic Ising model with multi-interactions. Therefore, it will be interesting to examine whether the already trained MLP can detect other models with untypical critical phenomena.
We study the phase transitions of three-dimensional (3D) classical O(3)O(3)O(3) model and two-dimensional (2D) classical XY model, as well as both the quantum phase transitions of 2D and 3D dimerized spin-1/2 antiferromagnets, using the technique of supervised neural network (NN). Moreover, unlike the conventional approaches commonly used in the literature, the training sets employed in our investigation are neither the theoretical nor the real configurations of the considered systems. Remarkably, with such an unconventional set up of the training stage in conjunction with some semiexperimental finite-size scaling formulas, the associated critical points determined by the NN method agree well with the established results in the literature. The outcomes obtained here imply that certain unconventional training strategies, like the one used in this study, are not only cost-effective in computation but are also applicable for a wide range of physical systems.
A neural network study of the phase transitions of the two-dimensional antiferromagnetic qqq-state Potts models on the square lattice
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