STEP: Spin Tensor Equivariant Potential for Data-Efficient Learning of Magnetic Potential Energy Surfaces

Abstract

Accurate and efficient modeling of magnetic potential energy surfaces remains challenging because spin-polarized first-principles calculations for diverse non-collinear spin-lattice configurations are computationally demanding. Here we introduce the Spin Tensor Equivariant Potential (STEP), a magnetic machine-learning interatomic potential that treats vector magnetic moments as continuous geometric degrees of freedom and embeds them in an equivariant representation. By coupling the central spin representation to its local spin-lattice environment through a Center-Environment Tensor Product, STEP introduces a physics-informed bias while preserving translational invariance and $\mathrm{SO}(3)$ equivariance and supporting feature-level time-reversal symmetrization. Learning-curve analysis on monolayer CrI$_3$ shows that STEP achieves pronounced data efficiency, with higher-order tensor channels and iterative center-environment couplings leading to steep learning curves for energy, force, and magnetic force errors. On public FeAl, CrN, and Fe benchmarks, STEP achieves competitive or improved accuracy compared with recent magnetic machine-learning potentials. Using a compact but representative CrI$_3$ dataset, STEP reproduces phonon dispersions and magnon spectra with high fidelity, capturing subtle anisotropic magnetic interactions. For Fe$_2$Mo$_3$O$_8$, STEP further provides a quantitative description of magnon--phonon hybridization and reproduces its characteristic magnon polaron dispersion. Finally, spin dynamics simulations driven by STEP yield Curie temperatures for monolayer CrI$_3$ and bcc Fe in good agreement with experiments. These results establish STEP as a physically informed, data-efficient, and scalable framework for modeling spin-lattice coupling, magnetic excitations, and finite-temperature magnetic behavior.