This study introduces time-reversal E(3)-equivariant neural network and SpinGNN++ framework for constructing a comprehensive interatomic potential for magnetic systems, encompassing spin-orbit coupling and noncollinear magnetic moments. SpinGNN++ integrates multitask spin equivariant neural network with explicit spin-lattice terms, including Heisenberg, Dzyaloshinskii-Moriya, Kitaev, single-ion anisotropy, and biquadratic interactions, and employs time-reversal equivariant neural network to learn high-order spin-lattice interactions using time-reversal E(3)-equivariant convolutions. To validate SpinGNN++, a complex magnetic model dataset is introduced as a benchmark and employed to demonstrate its capabilities. SpinGNN++ provides accurate descriptions of the complex spin-lattice coupling in monolayer CrI$_3$ and CrTe$_2$, achieving sub-meV errors. Importantly, it facilitates large-scale parallel spin-lattice dynamics, thereby enabling the exploration of associated properties, including the magnetic ground state and phase transition. Remarkably, SpinGNN++ identifies a new ferrimagnetic state as the ground magnetic state for monolayer CrTe2, thereby enriching its phase diagram and providing deeper insights into the distinct magnetic signals observed in various experiments.

翻译：本研究提出了时间反演E(3)等变神经网络与SpinGNN++框架，用于构建涵盖自旋-轨道耦合和非共线磁矩的磁性系统综合原子间势。SpinGNN++整合了多任务自旋等变神经网络与显式自旋-晶格项（包括海森堡、Dzyaloshinskii-Moriya、Kitaev、单离子各向异性和双二次相互作用），并采用时间反演等变神经网络通过时间反演E(3)等变卷积学习高阶自旋-晶格相互作用。为验证SpinGNN++，引入了一个复杂磁性模型数据集作为基准并用于展示其能力。SpinGNN++能精确描述单层CrI$_3$和CrTe$_2$中的复杂自旋-晶格耦合，实现亚meV误差。重要的是，该框架支持大规模并行自旋-晶格动力学，从而能够探索相关性质，包括磁基态和相变。值得注意的是，SpinGNN++识别出单层CrTe2中一种新的亚铁磁态作为基态磁结构，丰富其相图并为不同实验中观测到的独特磁信号提供了更深入的理解。