Representing magnetic environment: Magnetic descriptors for machine learning assisted spin dynamics

Machine learning (ML) based interatomic potential and force-field models have attracted enormous interest among researchers in quantum chemistry and physics. By accurately emulating the time-consuming quantum calculations for atomic forces, such ML models allow for large-scale and long time quantum molecular dynamics with desired quantum accuracy. The tremendous success of ML methods in quantum MD simulations has spurred similar approaches to multi-scale dynamical modeling of itinerant electron magnets which often exhibits highly complex spin textures such as vortices and skyrmions. A crucial requirement for a faithful ML model is to preserve symmetries of the original physical systems, which in the case of spin dynamics include the discrete lattice point group symmetry and spin rotation symmetry. Here we formulate a general theory of the magnetic descriptor for the spin field. Our approach focuses on the group-theoretical method that offers a systematic and rigorous methods to compute symmetry-invariant feature variables. Explicit implementation of magnetic descriptors is demonstrated for three types of itinerant magnets: the metallic spin glass, Kondo-lattice model, and spin-orbit-coupled chiral magnet.