Magnetic potential energy surface is crucial for understanding magnetic materials. This study introduces a time-reversal $E\left(3\right)$-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 $E\left(3\right)$-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.

中文翻译：

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磁性材料的物理信息时间反转等变神经网络势


磁势能面对于理解磁性材料至关重要。这项研究引入了时间反转 $E\left(3\right)$ -等变神经网络和基于物理的 SpinGNN++ 框架，用于构建磁系统的原子间势，包括自旋轨道耦合和非共线磁矩。 SpinGNN++ 将多任务自旋等变神经网络与显式自旋晶格项和时间反转等变神经网络集成，以使用时间反转学习高阶自旋晶格相互作用 $E\left(3\right)$ -等变卷积。引入复杂的磁模型数据集作为基准并用于展示其功能。 SpinGNN++ 提供了单层复杂自旋晶格耦合的准确描述 ${\mathrm{CrI}}_3$ 和 ${\mathrm{CrTe}}_2$ ，实现亚兆伏级误差并促进大规模并行自旋晶格动力学，从而能够探索相关特性，包括磁基态和相变。值得注意的是，SpinGNN++ 将不同的亚铁磁态识别为单层的基态 ${\mathrm{CrTe}}_2$ ，从而丰富了它的相图，并提供了对各种实验中观察到的不同磁信号的更深入的了解。