We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a “Shannon entropy” of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract-through a conventional finite-size scaling analysis with modest lattice sizes-the critical temperature with less than 1% error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.
We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a “Shannon entropy” of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract-through a conventional finite-size scaling analysis with modest lattice sizes-the critical temperature with less than 1% error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.
The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics.Here, we study how the minimum number of variables needed to accurately describe the important features of a data set-the intrinsic dimension (I d )-behaves in the vicinity of phase transitions.We employ state-of-the-art nearest-neighbors-based I d estimators to compute the I d of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless.For all the considered cases, we find that the I d uniquely characterizes the transition regime.The finite-size analysis of the I d allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent ν in the case of continuous transitions.For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods.Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space.
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.
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.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language
understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data,
phase recognition and study of critical properties of a system undergoing phase transition.
As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two
guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins.
The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention
to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model.
The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task,
we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model;
while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures.
The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical
temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into
subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the
proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature
in the thermodynamic limit, and the ν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional
Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a
Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field.
A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks
and learning tools. The most relevant codes are provided in the appendix.
We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow the exploration of different phases and symmetry-breaking, but they can distinguish phase-transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the "charge" correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the "autoencoder method," and we demonstrate that it too can be trained to capture phase transitions and critical points.
We assess the quality of a set of machine-learning (ML) techniques for the prediction of the phase diagrams of two frustrated 2D Ising models with competing interactions, that feature phase transitions characterized by critical manifolds rather than a single critical point. Using raw Monte Carlo spin configurations generated for random system parameters, we apply principal-component analysis (PCA) and auto-encoders to achieve dimensionality reduction, followed by clustering using the DBSCAN method and a support-vector machine classifier. The resulting estimates for the critical manifolds of both systems are in excellent agreement with available exact results, with the auto-encoders leading to quantitatively superior results. Notably, both the structure of the optimized auto-encoder as well its corresponding dimensionality-reduced and clustered space correlate with fundamental physical characteristics of the systems under consideration, which is useful when \emph{a priori} theoretical insight is unavailable.