Integrating deep-learning-based magnetic model and non-collinear spin-constrained method: methodology, implementation and application

We propose a non-collinear spin-constrained method that generates training data for deep-learning-based magnetic model, which provides a powerful tool for studying complex magnetic phenomena that requires large-scale simulations at the atomic level. First, we propose a basis-independent projection method for calculating atomic magnetic moments by applying a radial truncation to numerical atomic orbitals. A double-loop Lagrange multiplier method is utilized to ensure the satisfaction of constraint conditions while achieving accurate magnetic torque. The method is implemented in ABACUS with both plane wave basis and numerical atomic orbital basis. We benchmark the iron (Fe) systems and analyze differences from calculations with the plane wave basis and numerical atomic orbitals basis in describing magnetic energy barriers. Based on an automated workflow composed of first-principles calculations, magnetic model, active learning, and dynamics simulation, more than 30,000 first-principles data with the information of magnetic torque are generated to train a deep-learning-based magnetic model DeePSPIN for the Fe system. By utilizing the model in large-scale molecular dynamics simulations, we successfully predict Curie temperatures of α-Fe close to experimental values.