Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.
Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.
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 techniques have proved successful in identifying ordered phases of matter. However, it remains an open question how far they can contribute to the understanding of phases without broken symmetry, such as spin liquids. Here we demonstrate how a machine-learning approach can automatically learn the intricate phase diagram of a classical frustrated spin model. The method we employ is a support vector machine equipped with a tensorial kernel and a spectral graph analysis which admits its applicability in an effectively unsupervised context. Thanks to the interpretability of the machine we are able to infer, in closed form, both order parameter tensors of phases with broken symmetry, and the local constraints which signal an emergent gauge structure, and so characterize classical spin liquids. The method is applied to the classical XXZ model on the pyrochlore lattice where it distinguishes, among others, between a hidden biaxial spin-nematic phase and several different classical spin liquids. The results are in full agreement with a previous analysis by Taillefumier et al. [Phys. Rev. X 7, 041057 (2017)], but go further by providing a systematic hierarchy between disordered regimes, and establishing the physical relevance of the susceptibilities associated with the local constraints. Our work paves the way for the search of new orders and spin liquids in generic frustrated magnets.
Various types of multipolar order are ubiquitously found in frustrated spin systems, yet their identification is difficult and often causes confusion with spin liquids. In this work, by amalgamating the theory of orientational tensors and the strong interpretability of a particular machine-learning technique, the authors introduce an algorithm to detect the occurrence and the order parameter of hidden multipolar order and demonstrate its advantage over traditional methods. This approach provides a new scheme in the search for novel magnets as well as a comprehensive way to scrutinize spin liquid candidates.
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.
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.
Kitaev materials are promising materials for hosting quantum spin liquids and investigating the interplay of topological and symmetry-breaking phases.We use an unsupervised and interpretable machine-learning method, the tensorial-kernel support vector machine, to study the honeycomb Kitaev-model in a magnetic field.Our machine learns the global classical phase diagram and the associated analytical order parameters, including several distinct spin liquids, two exotic S 3 magnets, and two modulated S 3 × Z 3 magnets.We find that the extension of Kitaev spin liquids and a field-induced suppression of magnetic order already occur in the large-S limit, implying that critical parts of the physics of Kitaev materials can be understood at the classical level.Moreover, the two S 3 × Z 3 orders are induced by competition between Kitaev and spin liquids and feature a different type of spin-lattice entangled modulation, which requires a matrix description instead of scalar phase factors.Our work provides a direct instance of a machine detecting new phases and paves the way towards the development of automated tools to explore unsolved problems in many-body physics.