A Machine Learning study of the two-dimensional antiferromagnetic $q$-state Potts model on the square lattice

TL;DR

This work tackles the challenge of locating critical behavior in 2D antiferromagnetic $q$-state Potts models on the square lattice ($q=2$–$6$). It employs an ultra-simple multilayer perceptron trained only on two artificial stagger-type configurations, testing on Monte Carlo configurations to infer pseudo-critical temperatures via the output magnitude $R$, with $R=(1+1/\sqrt{2})/2$ marking $T_c(L)$. The key findings reproduce known physics: $q=2$ orders at finite $T$ with $T_c\approx 1.1346$, $q=3$ orders only at $T=0$, and $q=4$–$6$ remain disordered at all $T$; the method works across multiple $L$ and training choices and shows potential universality for detecting unusual critical phenomena. The results indicate that a minimal NN can capture complex phase behavior without using real spin configurations, offering a computationally efficient tool for exploring criticality in lattice models and potentially extending to other systems with untypical phase transitions.

Abstract

The critical phenomena of two-dimensional (2D) antiferromagnetic $q$-state Potts model on the square lattice with $q=2,3,4,5$ 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 $q=3$ model is critical only at zero temperature and $q=4,5,6$ 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.

Figures (10)

• Figure 1: A graphical representation for one of the MLP considered in this study. Here, the value of $N$, i.e. the number of each type of configurations included in the whole training set, is either 200 or 400. The figure is taken from Ref. Li25.
• Figure 2: The building units of the training set employed in this study. The value of $L_1$ can be varied. The figure is taken from Ref. Li25.
• Figure 3: 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 $L$ square area.
• Figure 4: $R$ as functions of $T$ for various $L$. The vertical solid and the horizontal dashed-dotted lines are the expected $T_c = 1.1346$ and $\frac{1+1/\sqrt{2}}{2}$, respectively. The results are determined by a MLP trained with a fixed $L_1 = 32$ using the building units of fig. \ref{['train']}.
• Figure 5: $R$ as functions of $T$ for various $L$. The vertical solid and the horizontal dashed-dotted lines are the expected $T_c = 1.1346$ and $\frac{1+1/\sqrt{2}}{2}$, respectively. For each considered $L$, the results are determined by a MLP trained with a fixed $L_1 = L$ using the building units of fig. \ref{['train']}.