Non-collinear magnetic atomic cluster expansion for iron

www.nature.com/npjcompumats ARTICLE OPEN Non-collinear magnetic atomic cluster expansion for iron Matteo Rinaldi 1✉ , Matous Mrovec1, Anton Bochkarev1, Yury Lysogorskiy1 and Ralf Drautz 1✉ The Atomic Cluster Expansion (ACE) provides a formally complete basis for the local atomic environment. ACE is not limited to representing energies as a function of atomic positions and chemical species, but can be generalized to vectorial or tensorial properties and to incorporate further degrees of freedom (DOF). This is crucial for magnetic materials with potential energy surfaces that depend on atomic positions and atomic magnetic moments simultaneously. In this work, we employ the ACE formalism to develop a non-collinear magnetic ACE parametrization for the prototypical magnetic element Fe. The model is trained on a broad range of collinear and non-collinear magnetic structures calculated using spin density functional theory. We demonstrate that the non-collinear magnetic ACE is able to reproduce not only ground state properties of various magnetic phases of Fe but also the magnetic and lattice excitations that are essential for a correct description of finite temperature behavior and properties of crystal defects. 1234567890():,; npj Computational Materials (2024)10:12 ; https://doi.org/10.1038/s41524-024-01196-8 INTRODUCTION Recent advancements of data-driven methods and machinelearned (ML) interatomic potentials have led to dramatically improved descriptions of the potential energy surface (PES) for many material systems. However, the incorporation of spin degrees of freedom (DOF), which are crucial to capture finite temperature phenomena in magnetic materials, has remained a challenging endeavor. In spin density functional theory (SDFT), magnetizaton emerges from the competition of magnetic exchange and band energy contributions1,2, where the energy required for reshuffling electrons in up and down spin channels depends on the local density of states (DOS). The bimodal DOS of iron in the body-centred crystal (bcc) structure affords large DOS values close to the Fermi level, leading to larger magnetic moments than in the face-centred cubic (fcc) structure with its more unimodal DOS that is lower at the Fermi level3,4. This intricate interplay between magnetic and atomic structure implies that multi-atom multi-spin interactions are necessary for capturing different magnetic and atomic structures in a single model. Unlike approaches that were derived from electronic structure theory and that seamlessly incorporate the complexity of magnetic interactions5–7, classical interatomic potentials needed to be supplemented via suitable interaction terms that mimic the quantum exchange interactions. The simplest possibility was to employ a classical Heisenberg Hamiltonian8, where the atomic spin operators are substituted by spin vectors and the exchange interactions are parameterized using first-principles calculations9. Such strategies have been adopted also in most current ML approaches for magnetic systems. Nikolov et al.10 augmented the spectral neighborhood analysis potential (SNAP) framework with a two-spin bi-linear Heisenberg model with atomic magnetic moment magnitudes being fixed and independent of the environment. A similar approach, where a neural network was trained to describe contributions to the Heisenberg Hamiltonian based on the local magnetic environment, was developed by Yu et al.11. However, this approach did not include information about the underlying lattice and treated the magnetic moments as unit vectors. Eckhoff et al.12 extended the formalism based on Behler-Parrinello symmetry functions13 in a framework that was limited to collinear configurations. Magnetic moments as additional DOF were incorporated by Novikov et al.14 in the moment tensor potential framework15. Even though the description was confined to collinear moments only, the magnetic moment tensor potential was able to reproduce a number of thermodynamic properties of bulk bcc Fe. Recently, Domina et al.16 extended the SNAP framework to deal with arbitrary vectorial fields and demonstrated its functionality by training to non-collinear spin configurations generated using a model Landau-Heisenberg Hamiltonian. In a follow-up work, Suzuki et al.17 showed that it is necessary to include higher-order spindependent partial spectra to discriminate configurations with different spin orientations and magnetic anisotropy. Finally, aiming at large-scale spin-lattice dynamics simulations, Chapman et al.18 added a neural network correction term to an embedded atom method potential augmented with a Heisenberg-Landau Hamiltonian. The model was successfully applied in finite temperature simulations of bulk Fe phases as well as complex defects. However, due to its simplicity, absolute errors were in some cases larger than a few tens of meV that are comparable to the fluctuations of exchange parameters with temperature. Thus, none of the existing magnetic ML approaches has so far succeeded in achieving a transferable and quantitatively accurate description of magnetic interactions suitable for modeling magnetism in different crystal structures. We present an explicit treatment of non-collinear magnetic DOF within the atomic cluster expansion (ACE)19,20, which provides a complete basis in the space of atomic environments19,21. Accurate, transferable and computationally efficient parameterizations of ACE have been developed for diverse bonding environments including bulk metallic systems as well as covalent molecules22–26. Thanks to ACE universality, additional scalar, vectorial or tensorial DOF can be incorporated seamlessly into ACE models20. Specifically for magnetic systems, ACE provides a body-ordered decomposition of combined atomic and magnetic PES in terms of a complete set of basis functions that depend on atomic and magnetic DOF. The inclusion of magnetic DOF requires an extension of the ACE equivariant basis such that any transformation of the relevant translation and rotation symmetry group Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, 44801 Bochum, Germany. ✉email: ; 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences M. Rinaldi et al. 1234567890():,; 2 Fig. 1 DFT energy vs volume for FM bcc and fcc. Constant magnetic moment energy-volume curves for FM bcc and fcc phases computed using constrained DFT. The black curve marks the ground state configurations without any applied constrain. The two minima for fcc correspond to the high- and low-spin magnetic configurations. acting on both atomic and magnetic spaces leaves the energy invariant. Magnetic ACE can therefore be considered as a generalization of most existing magnetic ML models as well as the classical spin-cluster expansion (SCE)27–31. In this work, we develop a non-collinear magnetic ACE parameterization for the prototypical magnetic element Fe. The mode (...truncated)