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 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% 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.
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 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 extend the Prometheus framework for unsupervised phase transition discovery from two-dimensional classical systems to three-dimensional classical systems and quantum many-body systems. Building upon preliminary observations from a 2D Ising model student abstract [Yee et al., 2026], we address two fundamental questions: (1) Does the framework scale to higher dimensions where exact solutions are unavailable? (2) Can it generalize to quantum phase transitions driven by quantum fluctuations rather than thermal fluctuations? For the 3D Ising model on lattices up to L=32, we achieve critical temperature detection within 0.01\% of literature values ($\Tc/J = 4.511 \pm 0.005$) and extract critical exponents with ≥70% accuracy, with statistical analysis correctly identifying the 3D Ising universality class (p=0.72). For quantum systems, we develop quantum-aware VAE (Q-VAE) architectures operating on complex-valued wavefunctions with fidelity-based loss functions, achieving 2\% accuracy in quantum critical point detection for the transverse field Ising model. For the disordered TFIM, we perform a consistency check of activated dynamical scaling $\ln ξ\sim |h - \hc|^{-ψ}$, extracting tunneling exponent ψ=0.48±0.08 consistent with theoretical predictions (ψ=0.5, Δχ2=12.3, p<0.001). This demonstrates that unsupervised learning can identify qualitatively different \emph{types} of critical behavior, serving as a consistency check on known IRFP physics.
The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.
Unsupervised Discovery of Intermediate Phase Order in the Frustrated J1-J2 Heisenberg Model via Prometheus Framework
The spin-1/2J1-J2 Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously applied to classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the J1-J2 phase diagram using a multi-scale approach. For L=4, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems (L=6,8), we introduce a reduced density matrix (RDM) based methodology using DMRG ground states, enabling scaling beyond the exponential barrier of full Hilbert space representation. Through dense parameter scans of J2/J1∈[0,1] and comprehensive latent space analysis, we identify the structure factor S(π,π) and S(π,0) as the dominant order parameters discovered by the VAE, with correlations exceeding ∣r∣>0.97. The RDM-VAE approach successfully captures the Néel-to-stripe crossover near J2/J1≈0.5--0.6, demonstrating that local quantum correlations encoded in reduced density matrices contain sufficient information for unsupervised phase discovery. This work establishes a scalable pathway for applying machine learning to frustrated quantum systems where full wavefunction access is computationally prohibitive.
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.