A Machine Learning study of the two-dimensional antiferromagnetic Ising model with nearest and next-to-nearest interactions on the triangular lattice

TL;DR

This study applies a minimal multilayer perceptron (MLP) to the 2D antiferromagnetic Ising model on a triangular lattice with nearest and next-nearest couplings $J_1$ and $J_2$, seeking critical temperatures and transition orders for $J_2/J_1 = 0.1, 0.5, 1.0$. Remarkably, the MLP is trained only on two artificially constructed staggered configurations, requiring no real spin configurations, yet it accurately detects $T_c$ and identifies first-order transitions across all three parameter sets, including a weak first-order case at $J_2/J_1=0.5$. The method yields clear signatures in the MLP output magnitude $R$ and aligns with established results (e.g., Ras05), while offering substantial speedups and potential universality for other models with complex ground states. The work suggests broad applicability to models with untypical phase transitions and nontrivial ground states, and it raises questions for extending the approach to additional frustrated systems and Potts-like variants.

Abstract

We study the phase transitions of the two-dimensional 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 (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 $J_2/J_1$ 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 $J_2/J_1 = 0.1, 0.5$ 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 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.

Figures (15)

• Figure 1: The antiferromagnetic $J_1$-$J_2$ Ising model on the triangular lattice studied here. The solid and the dashed lines represent the nearest coupling $J_1$ and next-to nearest coupling $J_2$, respectively. Only one $J_2$ is shown.
• Figure 2: Representation for one of the MLPs used in this study. $L_1$ and $L_2$ are the linear system sizes for the parallelogram covering a triangular lattice having $L_1 \times L_2$ sites. $N$ can be any number and here we use $N=200$ or $N=400$. This figure is drawn based on the MLP of Ref. Tan20.1.
• Figure 3: The building blocks of the training set employed in this study. The value of $L_1$ can be varied.
• Figure 4: The construction of a configuration for the testing set related to a given linear box size $L_1$ from a spin configuration on a $L$ by $L$ parallelogram.
• Figure 5: The initial configurations (on a 6 by 6 parallelogram), namely stagger-start (top left panel), uniform-start (top right panel), and randomness-start (down left panel) used to begin the MC simulations. Arrows which point upward (downward) stand for spins with their values being 1 (-1).