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.
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.
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 Tc. 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 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~, 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) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(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.
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.
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
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.
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.