  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.
