Integrating Deep-Learning-Based Magnetic Model and Non-Collinear Spin-Constrained Method: Methodology, Implementation and Application

TL;DR

The paper addresses the challenge of modeling complex magnetic phenomena at large scales by integrating a non-collinear spin-constrained DFT framework with a deep-learning magnetic model. It introduces a basis-independent projection method using smooth modulation orbitals and a double-loop Lagrange multiplier approach to generate high-quality training data for DeePSPIN within ABACUS, usable with both plane-wave and NAO bases. An automated workflow, driven by DPGEN active learning, produces thousands of first-principles data, enabling DeePSPIN to reproduce energetics and magnetic torques and to run large-scale MD that captures the ferromagnetic–paramagnetic transition in Fe near the experimental Curie temperature. Validation includes rigorous finite-difference checks and comparisons of magnetic energy surfaces for BCC-Fe and FCC-Fe, demonstrating robustness across basis sets and non-collinear spin states. The work provides a scalable, automated pipeline for AI-driven magnetic materials simulation, including open data and toolchains to facilitate further research and applications.

Abstract

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 alpha-Fe close to experimental values.

Figures (11)

• Figure 1: Radial functions (RFs) of the first-$\zeta$ numerical atomic orbital of Fe before and after modulation. (a) The original and modulated RFs for $p$ orbitals, where the original numerical atomic orbitals have a cutoff radius $r_c$ of 6.0 Bohr, as well as the modulated orbital with the modification radius $r_m$ as 1.0, 2.0 and 4.0 Bohr. "p-1.0" represents the $p$ orbital modulated by Eq. \ref{['eq:orb_mod']} with $r_m=1.0$ Bohr. (b) The original and modulated RFs for $d$ orbitals.
• Figure 2: Estimation of BCC-Fe atomic magnetic moments as a function of modulation radius $r_m$. The modulated orbital projection algorithm is employed with DZP and TZDP basis sets with cutoff radius $r_c$ ranging from 6.0 to 10.0 Bohr, to demonstrate the impact of different NAO basis sets on atomic magnetic moments. Where (a) is for the PW basis set in the FM magnetic configuration, with the reference value being the total magnetization per atom (TMAG); (b) is for the NAO basis set in the FM magnetic configuration, with the reference value being the TMAG; (c) is for the PW basis set in the AFM magnetic configuration, with the reference value being the absolute magnetization per atom (AMAG); and (d) is for the LCAO basis set in the AFM magnetic configuration, with the reference value being the atomic magnetization from Mulliken charge.
• Figure 3: The total energy and magnetic torque of BCC-Fe as function of modulation radius. (a) The total energy of BCC-Fe calculated at different modulation radius by using DZP basis ($r_c=7.0~\text{Bohr}$). The labels represent the angle of magnetic moment between two nearest-neighbor Fe atoms in BCC-Fe, while the magnetic moment magnitudes are set to the corresponding FM ground-state values for each $r_m$, as indicated by the black pentagrams in (b). (b) shows the corresponding magnetic torque $|\boldsymbol{\lambda}|$ of BCC-Fe calculated at different modulation radius.
• Figure 4: Finite difference tests. (a) Finite difference tests for the atomic force in BCC Fe$_{16}$ with a perturbation step of 0.01 Bohr along z-direction. (b) Finite difference tests for the magnetic torque in binary alloy FePt with a perturbation step of 0.1 $\mu_B$. (c) Finite difference tests for the cell stress in ternary alloy NiMnTi with a perturbation step 0.0001. "11" refers to the component of stress matrix $\sigma_{11}$. In (a-c), "Analytic" represents the value calculated from the formula, while "Numerical" is the value determined from the finite difference method.
• Figure 5: Comparison between the unconstrained and constrained calculations. (a,b) plot the magnetic force $\lambda=|\boldsymbol\lambda|$ as a function of the cell volume per atom for BCC-Fe (a) and FCC-Fe (b). (c,d) The solid lines show the total energy per atom as a function of the cell volume per atom for BCC-Fe (c) and FCC-Fe (d). Here "DZP-8au-v2.1" represents the v2.1 NAOs calculations based on the DZP orbitals with the 8 Bohr cutoff. The gray histograms represent the atomic magnetic moment $m$ after fully unconstrained self-consistent calculations. The dotted lines show the cDFT energy with certain constrains.