Revealing the Phase Diagram of Kitaev Materials by Machine Learning: Cooperation and Competition between Spin Liquids

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 classical honeycomb Kitaev-$\Gamma$ model in a magnetic field. Our machine learns the global phase diagram and the associated analytical order parameters, including several distinct spin liquids, two exotic $S_3$ magnets, and two modulated $S_3 \times Z_3$ 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 $S_3 \times Z_3$ orders are induced by competition between Kitaev and $\Gamma$ spin liquids and feature a previously unknown type of spin-lattice entangled modulation, which requires a matrix description instead of scalar phase factors. Our work provides the first 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.

Probing Hidden Spin Order with Interpretable Machine Learning

Machine learning shows promise for improving our understanding of many-body problems. Tackling an unsolved problem, or classifying intricate phases, remains however a daunting task. We introduce a generic protocol to probe and identify multipolar spin order, and extract the analytical form of the tensorial order parameter up to rank 6. Moreover, we find that our approach yields reliable results already for a modest amount of training data and without knowledge of the exact phase boundary. Our approach may prove useful for identifying novel spin order and ruling out spurious spin liquid candidates.