Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman V. Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, Alexandre Dauphin
Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.
Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.
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.
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.
The classification of phase transitions is a central and challenging task in condensed matter physics. Typically, it relies on the identification of order parameters and the analysis of singularities in the free energy and its derivatives. Here, we propose an alternative framework to identify quantum phase transitions, employing both unsupervised and supervised machine learning techniques. Using the axial next-nearest-neighbor Ising (ANNNI) model as a benchmark, we show how unsupervised learning can detect three phases (ferromagnetic, paramagnetic, and a cluster of the antiphase with the floating phase) as well as two distinct regions within the paramagnetic phase. Employing supervised learning we show that transfer learning becomes possible: a machine trained only with nearest-neighbor interactions can learn to identify a new type of phase occurring when next-nearest-neighbor interactions are introduced. All our results rely on few- and low-dimensional input data (up to twelve lattice sites), thus providing a computational friendly and general framework for the study of phase transitions in many-body systems.
Machine learning is capable of discriminating phases of matter, and finding associated phase transitions, directly from large data sets of raw state configurations. In the context of condensed matter physics, most progress in the field of supervised learning has come from employing neural networks as classifiers. Although very powerful, such algorithms suffer from a lack of interpretability, which is usually desired in scientific applications in order to associate learned features with physical phenomena. In this paper, we explore support vector machines (SVMs), which are a class of supervised kernel methods that provide interpretable decision functions. We find that SVMs can learn the mathematical form of physical discriminators, such as order parameters and Hamiltonian constraints, for a set of two-dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. The ability of SVMs to provide interpretable classification highlights their potential for automating feature detection in both synthetic and experimental data sets for condensed matter and other many-body systems.
Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.
Abstract Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.