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arXiv:2306.16822

Semi-supervised learning of order parameter in 2D Ising and XY models using Conditional Variational Autoencoders

adwait naravane, Nilmani Mathur A

Published Jun 29, 2023• 0 citations
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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 $\mathbb{Z}_2$ 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.

DOI
10.48550/arxiv.2306.16822

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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.
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.
Machine learning methods have been important in the study of phase transitions. Unsupervised methods are particularly attractive because they do not require prior knowledge of the existence of a phase transition. In this work we focus on the constant magnetization Ising model in two (2D) and three (3D) dimensions. While there have been many studies using machine learning for the critical behavior of these systems, we are not aware of any studies for the phase diagram at off-critical magnetizations below the critical temperature. Previous work has used the raw spins as the input feature. We show that a more robust input feature is the local affinity, where the value of the feature at each site is determined by the spin and its neighbors. When coupled with a variational autoencoder, the method is able to predict the phase behavior of the 2D and 3D Ising models (including the critical exponent β) in quantitative agreement with conventional simulations. The choice of activation functions in the autoencoder is crucial, and this requires physical insight into the nature of the phase transition. The method is general and can be applied to any lattice or off-lattice system.