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arXiv:2603.20157 · APS Open Science

Detecting the three-dimensional Ising model phase transition with a ground-state-trained autoencoder

A. M. Abuali, David A. Clarke, Morten Hjorth-Jensen, Ioannis Konstantinidis, Claudia Ratti, Jianyi Yang

Published Jul 16, 2026• 0 citations
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Abstract

We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the three-dimensional (3D) Ising model. A 3D convolutional autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature T c = 4.5128 ( 58 ) , in excellent agreement with the known value. Additionally, we obtain an estimate of the correlation-length critical exponent, ν = 0.63 ( 27 ) , also consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.

DOI
10.1103/h8n4-94g4

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We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the three-dimensional (3D) Ising model. A 3D convolutional autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature T c = 4.5128 ( 58 ) , in excellent agreement with the known value. Additionally, we obtain an estimate of the correlation-length critical exponent, ν = 0.63 ( 27 ) , also consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.
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We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.
Abstract We present Prometheus, an unsupervised machine learning framework for discovering phase transitions and extracting critical exponents in three-dimensional spin systems. Building upon our previous work on the two-dimensional Ising model, we extend the variational autoencoder (VAE) approach to three-dimensional systems, demonstrating that the learned latent representations capture the essential physics of the ferromagnetic-paramagnetic phase transition. Our 3D convolutional VAE architecture achieves ≥70%\geq 70\%≥70% accuracy in critical exponent extraction for the 3D Ising model, with the critical temperature $\Tc = 4.511 \pm 0.005$ in excellent agreement with the theoretical value $\Tc/J = 4.5115$. The framework incorporates a novel quantum discovery mechanism that enables systematic exploration of parameter spaces and automated detection of novel phase transition phenomena. We validate our approach through extensive finite-size scaling analysis, bootstrap confidence intervals, and comparison with established Monte Carlo results. The Prometheus framework provides a powerful tool for unsupervised exploration of phase diagrams in complex many-body systems where analytical solutions are unavailable.
In this study, we computed three critical exponents (α,β,γα, β, γα,β,γ) for the 3D Ising model with Metropolis Algorithm using Finite-Size Scaling Analysis on six cube length scales (L=20,30,40,60,80,90), and performed a supervised Deep Learning (DL) approach (3D Convolutional Neural Network or CNN) to train a neural network on specific conformations of spin states. We find one can effectively reduce the information in thermodynamic ensemble-averaged quantities vs. reduced temperature t (magnetization per spin (t)(t)(t), specific heat per spin (t)(t)(t), magnetic susceptibility per spin <χ>(t)<χ>(t)<χ>(t)) to \textit{six} latent classes. We also demonstrate our CNN on a subset of L=20 conformations and achieve a train/test accuracy of 0.92 and 0.6875, respectively. However, more work remains to be done to quantify the feasibility of computing critical exponents from the output class labels (binned m,c,χm, c, χm,c,χ) from this approach and interpreting the results from DL models trained on systems in Condensed Matter Physics in general.