Abstract We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial D 3h spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading 3 × 3 correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.
Abstract We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial D 3h spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading 3 × 3 correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.
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.
We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine, to investigate the low-temperature classical phase diagram of a generalized Heisenberg-Kitaev-$\mathrm{\ensuremath{\Gamma}}$ ($J\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated ${S}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ phase, which emerges through the competition between the Kitaev and $\mathrm{\ensuremath{\Gamma}}$ spin liquids, against Heisenberg interactions. The results imply that, in the restricted parameter space spanned by the three primary exchange interactions---J, K, and $\mathrm{\ensuremath{\Gamma}}$, the representative Kitaev material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ lies close to the boundaries of several phases, including a simple ferromagnet, the unconventional ${S}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$, and nested zigzag-stripy magnets. A zigzag order is stabilized by a finite ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ and/or J3 term, whereas the four magnetic orders may compete in particular if ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ is antiferromagnetic.
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.
Classical machine learning has proven remarkably useful in post-processing quantum data, yet typical learning algorithms often require prior training to be effective. In this work, we employ a tensorial kernel support vector machine (TK-SVM) to analyze experimental data produced by trapped-ion quantum computers. This unsupervised method benefits from directly interpretable training parameters, allowing it to identify the non-trivial string-order characterizing symmetry-protected topological (SPT) phases. We apply our technique to two examples: a spin-1/2 model and a spin-1 model, featuring the cluster state and the AKLT state as paradigmatic instances of SPT order, respectively. Using matrix product states, we generate a family of quantum circuits that host a trivial phase and an SPT phase, with a sharp phase transition between them. For the spin-1 case, we implement these circuits on two distinct trapped-ion machines based on qubits and qutrits. Our results demonstrate that the TK-SVM method successfully distinguishes the two phases across all noisy experimental datasets, highlighting its robustness and effectiveness in quantum data interpretation.
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.
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.