Abstract Machine learning techniques have been widely used in the study of strongly correlated systems in recent years. Here, we review some applications to classical and quantum many-body systems and present results from an unsupervised machine learning technique, the principal component analysis, employed to identify the finite-temperature phase transition of the three-dimensional Fermi-Hubbard model to the antiferromagnetically ordered state. We find that this linear method can capture the phase transition as well as other more complicated and nonlinear counterparts.
Abstract Machine learning techniques have been widely used in the study of strongly correlated systems in recent years. Here, we review some applications to classical and quantum many-body systems and present results from an unsupervised machine learning technique, the principal component analysis, employed to identify the finite-temperature phase transition of the three-dimensional Fermi-Hubbard model to the antiferromagnetically ordered state. We find that this linear method can capture the phase transition as well as other more complicated and nonlinear counterparts.
We apply various unsupervised machine learning methods for phase classification to investigate the finite-temperature phase diagram of the spinless Falicov-Kimball model in two dimensions. Using only particle occupation snapshots from Monte Carlo simulations as input, each technique, including a straightforward classification based on principal component analysis (PCA), successfully identifies the phase boundary between ordered and disordered phases, independent of the type of phase transition. Remarkably, these techniques also distinguish between the weakly localized and Anderson-localized regimes within the disordered phase, accurately identifying their crossover, which is a challenging task for standard methods. Among the machine learning approaches used, PCA based analysis outperforms more complex methods, such as neural network predictors and autoencoders. These results underscore the effectiveness of simple unsupervised techniques in examining phase transitions and electron localization in complex correlated systems.
We use determinant quantum Monte Carlo (DQMC), in combination with the principal component analysis (PCA) approach to unsupervised learning, to extract information about phase transitions in several of the most fundamental Hamiltonians describing strongly correlated materials. We first explore the zero-temperature antiferromagnet to singlet transition in the periodic Anderson model, the Mott insulating transition in the Hubbard model on a honeycomb lattice, and the magnetic transition in the 1/6-filled Lieb lattice. We then discuss the prospects for learning finite temperature superconducting transitions in the attractive Hubbard model, for which there is no sign problem. Finally, we investigate finite temperature charge density wave (CDW) transitions in the Holstein model, where the electrons are coupled to phonon degrees of freedom, and carry out a finite size scaling analysis to determine Tc. We examine the different behaviors associated with Hubbard-Stratonovich auxiliary field configurations on both the entire space-time lattice and on a single imaginary time slice, or other quantities, such as equal-time Green's and pair-pair correlation functions.
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 employ several unsupervised machine learning techniques, including autoencoders, random trees embedding, and t-distributed stochastic neighboring ensemble (t-SNE), to reduce the dimensionality of, and therefore classify, raw (auxiliary) spin configurations generated, through Monte Carlo simulations of small clusters, for the Ising and Fermi-Hubbard models at finite temperatures. Results from a convolutional autoencoder for the three-dimensional Ising model can be shown to produce the magnetization and the susceptibility as a function of temperature with a high degree of accuracy. Quantum fluctuations distort this picture and prevent us from making such connections between the output of the autoencoder and physical observables for the Hubbard model. However, we are able to define an indicator based on the output of the t-SNE algorithm that shows a near perfect agreement with the antiferromagnetic structure factor of the model in two and three spatial dimensions in the weak-coupling regime. t-SNE also predicts a transition to the canted antiferromagnetic phase for the three-dimensional model when a strong magnetic field is present. We show that these techniques cannot be expected to work away from half filling when the "sign problem" in quantum Monte Carlo simulations is present.
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 learning is a discipline of machine learning which aims at discovering patterns in large data sets or classifying the data into several categories without being trained explicitly. We show that unsupervised learning techniques can be readily used to identify phases and phases transitions of many-body systems. Starting with raw spin configurations of a prototypical Ising model, we use principal component analysis to extract relevant low-dimensional representations of the original data and use clustering analysis to identify distinct phases in the feature space. This approach successfully finds physical concepts such as the order parameter and structure factor to be indicators of a phase transition. We discuss the future prospects of discovering more complex phases and phase transitions using unsupervised learning techniques.