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arXiv:2005.13742 · Scientific Reports

Deep learning on the 2-dimensional Ising model to extract the crossover region with a variational autoencoder

Nicholas Walker, Ka-Ming Tam, Mark Jarrell

Published Aug 3, 2020• 41 citations
arXivHTMLPDF

Abstract

The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the configurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available.

DOI
10.1038/s41598-020-69848-5

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The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the configurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available.
The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the configurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available.
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
We investigate the application of deep learning techniques employing the conditional variational autoencoders for semi-supervised learning of latent parameters to describe phase transition in the two-dimensional (2D) ferromagnetic Ising model and the two-dimensional XY model. For both models, we utilize spin configurations generated using the Wolff algorithms below and above the critical temperatures. For the 2D Ising model we find the latent parameter of conditional variational autoencoders is correlated to the known order parameter of magnetization more efficiently than their correspondence in variational autoencoders used previously. It can also clearly identify the restoration of the Z2\mathbb{Z}_2Z2​ symmetry beyond the critical point. The critical temperature extracted from the latent parameter at larger lattices are found to be approaching its correct value. Similarly, for the 2D XY model, we find our chosen network with the latent representation of conditional variational autoencoders is equally capable of separating the two phases between the high and low temperatures, again at the correct critical temperature with reasonable accuracy. Together these results show that the latent representation of conditional variational autoencoders can be employed efficiently to identify the phases of condensed matter systems, without their prior knowledge.
We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We use spin configurations produced for the 2-dimensional ferromagnetic Ising model in zero external magnetic field. We study numerically the relation between one latent dimension extracted from the autoencoder to the critical temperature TcT_cTc​. The 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 Z2\mathbb{Z}_2Z2​ symmetry as the temperature increases. 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 dimension z~{\tilde z}z~, which enables us to predict the critical temperature. We show that one can build the latent susceptibility and use it to quantify the value of the critical temperature Tc(L)T_c(L)Tc​(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L)T_c(L)Tc​(L) extrapolates to the known theoretical value as L→∞L \to \inftyL→∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision.
Abstract We present an unsupervised machine learning framework for discovering phase transitions and order parameters in the two-dimensional Ising model without prior knowledge of the underlying physics. Using convolutional beta-variational autoencoders (β-VAE), we learn compressed representations that naturally separate ordered and disordered phases. Our physics-informed training procedures incorporate data augmentation respecting Ising symmetries, cosine annealing optimization, and progressive temperature curriculum learning. Experimental validation on Monte Carlo simulated configurations demonstrates automatic order parameter discovery with correlation ρ = 0.85 ± 0.04 to theoretical magnetization (95% CI [0.82, 0.88]) and critical temperature detection within 0.27% of the Onsager solution (p < 0.001). The method achieves 89% improvement over principal component analysis and 124% improvement over t-distributed stochastic neighbor embedding. Latent representations exhibit clear phase separation with physics consistency score S = 0.88 ± 0.03. Ablation studies reveal optimal performance at β = 1.0, balancing reconstruction fidelity with disentanglement. This framework enables automated identification of emergent phenomena in complex many-body systems, advancing machine learning applications in condensed matter physics.
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.