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Journal of Physics Communications

Deep learning study of nonequilibrium phase transitions in ferromagnetic systems under effective interactions

Dagne Wordofa Tola, Mulugeta Bekele

Published Aug 21, 2026• 0 citations
Publisher

Abstract

Abstract Phase transitions play a central role in statistical physics and are well understood in systems at thermal equilibrium. However, their nonequilibrium counterparts remain less explored because of the lack of a unified theoretical framework. In this study, we examine the potential of deep learning, specially convolutional neural networks (CNN), to detect nonequilibrium phase transitions (NEPT) in a 2D Ising model-like square lattice ferromagnetic system under effective parameter h . Without this parameter ( h = 0 ), we generate equilibrium spin configurations using the standard Metropolis algorithm as usual. With h ≠ 0 , we generate nonequilibrium spin configurations using an effective Glauber update rule which breaks detailed balance. While the former is used to train CNN, the later is not subsequently described by the usual Boltzmann-Gibbs distribution rather used to test the model. The study shows that the trained CNN can distinguish between ordered and disordered nonequilibrium steady states and accurately predict the critical temperature of the system undergoing NEPT. Finite size scaling analysis of the model’s output (perdition) reproduces critical exponents that agree with theoretical expectations, indicating that this method provides a strong, data driven approach for examining nonequilibrium critical phenomena.

DOI
10.1088/2399-6528/ae9d14

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Abstract Phase transitions play a central role in statistical physics and are well understood in systems at thermal equilibrium. However, their nonequilibrium counterparts remain less explored because of the lack of a unified theoretical framework. In this study, we examine the potential of deep learning, specially convolutional neural networks (CNN), to detect nonequilibrium phase transitions (NEPT) in a 2D Ising model-like square lattice ferromagnetic system under effective parameter h . Without this parameter ( h = 0 ), we generate equilibrium spin configurations using the standard Metropolis algorithm as usual. With h ≠ 0 , we generate nonequilibrium spin configurations using an effective Glauber update rule which breaks detailed balance. While the former is used to train CNN, the later is not subsequently described by the usual Boltzmann-Gibbs distribution rather used to test the model. The study shows that the trained CNN can distinguish between ordered and disordered nonequilibrium steady states and accurately predict the critical temperature of the system undergoing NEPT. Finite size scaling analysis of the model’s output (perdition) reproduces critical exponents that agree with theoretical expectations, indicating that this method provides a strong, data driven approach for examining nonequilibrium critical phenomena.
This paper presents the investigation of convolutional neural network (CNN) prediction successfully recognizing the temperature of the nonequilibrium phase transitions in two-dimensional (2D) Ising spins on a square lattice. The model uses image snapshots of ferromagnetic 2D spin configurations as an input shape to provide the average output predictions. By considering supervised machine learning techniques, we perform Metropolis Monte Carlo (MC) simulations to generate the configurations. In the equilibrium Ising model, the Metropolis algorithm respects detailed balance condition (DBC), while its nonequilibrium version violates DBC. Violating the DBC of the algorithm is characterized by a parameter −8<ε<8. We find the exact result of the transition temperature Tc(ε) in terms of ε. If we set ε=0, the usual single spin-flip algorithm can be restored, and the equilibrium configurations generated with such a set up are used to train our model. For ε≠0, the system attains the nonequilibrium steady states (NESS), and the modified algorithm generates NESS configurations (test dataset). The trained model is successfully tested on the test dataset. Our result shows that CNN can determine Tc(ε≠0) for various ε values, consistent with the exact result.
Quantum spin networks represent systems in which quantum spins are arranged on a topological lattice, where the nature of spin interactions and the influence of external magnetic fields can lead to complex collective behaviors. The Hamiltonian governing such systems determines the energy landscape and phase transitions under various thermodynamic conditions. In this paper, we focus on the two-dimensional Ising spin model with periodic boundary conditions subjected to a uniform external magnetic field. Initially, we employ the Metropolis Monte Carlo (MP-MN) algorithm to simulate the system and identify its phase transitions at different magnetic field strengths. The emergence of ordered and disordered phases in response to thermal fluctuations and magnetic interactions is systematically analyzed. Subsequently, we explore the application of convolutional neural networks (CNNs), a powerful class of deep learning models, to detect and classify the phases of the Ising model based on spin configurations generated at a fixed temperature. The CNN is trained using labeled data representing different magnetic field values, and its performance in phase prediction is quantitatively evaluated. The results demonstrate that CNNs can successfully learn complex spin patterns and provide accurate classification of spin phases, highlighting their potential for analyzing phase transitions in statistical physics models.
Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.
Quantum spin networks represent configurations of spins arranged on a topological lattice, where the spin interactions are governed by the system's Hamiltonian. These networks are critical for understanding magnetic materials, as the arrangement of spins and the type of interaction between neighboring spins determine the macroscopic behavior of the system. The behavior of these systems is further influenced by the presence of external magnetic fields. In this paper, we first investigate the various phases of the two-dimensional Ising lattice with periodic boundary conditions under the influence of a uniform external magnetic field. The exploration of these phases is performed using the Metropolis Monte Carlo (MP-MN) algorithm, a well-established statistical method for simulating spin systems. Subsequently, we explore the potential of deep learning, specifically convolutional neural networks (CNN), in identifying and predicting these phases of spin lattices. The CNN's ability to classify different phases of the two-dimensional Ising model in the presence of a homogeneous magnetic field at a constant temperature is examined. The study aims to demonstrate how machine learning models, particularly CNNs, can effectively detect phase transitions and predict the system's behavior, which traditionally requires extensive computational methods. Finally, the performance of the CNN algorithm is evaluated by assessing its accuracy in predicting different phases of the Ising model.
Abstract We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature T c . The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ 2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z ̃ , which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature T c ( L ) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that T c ( L ) extrapolates to the known theoretical value as L → ∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract
Abstract Employing a deep convolutional neural network (deep CNN) trained on spin configurations of the 2D Ising model and the temperatures, we examine whether the deep CNN can detect the phase transition of the 2D qqq-state Potts model. To this end, we generate binarized images of spin configurations of the qqq-state Potts model (q≥3q\ge 3q≥3) by replacing the spin variables {0,1,…,⌊q/2⌋−1}\{0,1,\ldots,\lfloor q/2\rfloor-1\}{0,1,…,⌊q/2⌋−1} and {⌊q/2⌋,…,q−1}\{\lfloor q/2\rfloor,\ldots,q-1\}{⌊q/2⌋,…,q−1} with {0}\{0\}{0} and {1}\{1\}{1}, respectively. Then, we input these images to the trained CNN to output the predicted temperatures. The binarized images of the qqq-state Potts model are entirely different from Ising spin configurations, particularly at the transition temperature. Moreover, our CNN model is not trained on the information about whether phases are ordered/disordered but is naively trained by Ising spin configurations labeled with temperatures at which they are generated. Nevertheless, the deep CNN can detect the transition point with high accuracy, regardless of the type of transition. We also find that, in the high-temperature region, the CNN outputs the temperature based on the internal energy, whereas, in the low-temperature region, the output depends on the magnetization and possibly the internal energy as well. However, in the vicinity of the transition point, the CNN may use more general factors to detect the transition point.