Physics-informed time-reversal equivariant neural network potential for magnetic materials

Magnetic potential energy surface is crucial for understanding magnetic materials. This study introduces a time-reversal $E(3)$-equivariant neural network and physics-informed SpinGNN++ framework for constructing interatomic potentials for magnetic systems, encompassing spin-orbit coupling and noncollinear magnetic moments. SpinGNN++ integrates multitask spin equivariant neural network with explicit spin-lattice terms and time-reversal equivariant neural network to learn high-order spin-lattice interactions using time-reversal $E(3)$-equivariant convolutions. A complex magnetic model data set is introduced as a benchmark and employed to demonstrate its capabilities. SpinGNN++ provides accurate descriptions of the complex spin-lattice coupling in monolayer ${\mathrm{CrI}}_{3}$ and ${\mathrm{CrTe}}_{2}$, achieving sub-meV errors and facilitates large-scale parallel spin-lattice dynamics, thereby enabling the exploration of associated properties, including magnetic ground state and phase transition. Remarkably, SpinGNN++ identifies a differentferrimagnetic state as the ground state for monolayer ${\mathrm{CrTe}}_{2}$, thereby enriching its phase diagram and providing deeper insights into the distinct magnetic signals observed in various experiments.