The Spin Tensor Equivariant Potential (STEP), a magnetic machine-learning interatomic potential that treats vector magnetic moments as continuous geometric degrees of freedom and embeds them in an equivariant representation, is introduced.
Abstract
Accurate and efficient modeling of magnetic potential energy surfaces remains challenging because spin-polarized first-principles calculations for diverse non-collinear spin-lattice configurations are computationally demanding. Here we introduce the Spin Tensor Equivariant Potential (STEP), a magnetic machine-learning interatomic potential that treats vector magnetic moments as continuous geometric degrees of freedom and embeds them in an equivariant representation. By coupling the central spin representation to its local spin-lattice environment through a Center-Environment Tensor Product, STEP introduces a physics-informed bias while preserving translational invariance and $\mathrm{SO}(3)$ equivariance and supporting feature-level time-reversal symmetrization. Learning-curve analysis on monolayer CrI$_3$ shows that STEP achieves pronounced data efficiency, with higher-order tensor channels and iterative center-environment couplings leading to steep learning curves for energy, force, and magnetic force errors. On public FeAl, CrN, and Fe benchmarks, STEP achieves competitive or improved accuracy compared with recent magnetic machine-learning potentials. Using a compact but representative CrI$_3$ dataset, STEP reproduces phonon dispersions and magnon spectra with high fidelity, capturing subtle anisotropic magnetic interactions. For Fe$_2$Mo$_3$O$_8$, STEP further provides a quantitative description of magnon--phonon hybridization and reproduces its characteristic magnon polaron dispersion. Finally, spin dynamics simulations driven by STEP yield Curie temperatures for monolayer CrI$_3$ and bcc Fe in good agreement with experiments. These results establish STEP as a physically informed, data-efficient, and scalable framework for modeling spin-lattice coupling, magnetic excitations, and finite-temperature magnetic behavior.
Magnetic materials exhibit an intricate coupling between atomic structure and spin degrees of freedom, posing a fundamental challenge for atomistic simulations across experimentally relevant length and time scales. Here we introduce HotPP-Spin, a spin-dependent extension of HotPP for magnetic machine learning interatomic potentials, built on Cartesian tensor equivariant message passing. Atomic magnetic moments are treated as explicit axial-vector degrees of freedom, while spatial-inversion and time-reversal parities are propagated through the tensor couplings. This construction provides a unified representation of exchange-dominated and spin-orbit-induced interactions without imposing predefined analytical interaction forms. A scalar spin-dependent potential energy surface yields energy-conserving atomic forces and magnetic effective fields through differentiation. Benchmarks spanning collinear magnetism, noncollinear magnetism, and spin-orbit-coupling-induced magnetic anisotropy show that HotPP-Spin accurately describes magnetic energy landscapes, magnetic forces, and magnetic-order-dependent energy-volume relations within the same general framework. For H-phase monolayer VSe\(_2\), stochastic spin-dynamics simulations using the learned magnetic effective fields locate the finite-size magnetic ordering crossover at 415--435~K, in close numerical agreement with the reported experimental value of \(418.5\pm7.8\)~K. These results establish Cartesian tensor message passing as a general route for connecting first-principles magnetic energetics with large-scale atomistic simulations of coupled structural and spin phenomena.
Junjie Wang, Yijie Zhu, Zhongwei Zhang et al.· 0 citations
A novel anisotropic machine learning CG potential is introduced that extends the point particle representation of atomic nuclei to massive ellipsoidal beads with orientation-dependent features, enabling the learning of energies, forces, and torques directly from atomistic data.
Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.
Equivariant graph neural networks encode the E(3) symmetries that molecular force fields must obey, yet they are often trained against geometric objectives whose own symmetry is rarely examined. We develop a symmetry-sensitive diagnostic protocol that combines rigid-transform invariance and equivariance metrics, a parity analysis of graph-level targets, an auxiliary-head predictability measure, and torsion and mirror perturbation tests. It separates three effects that bulk error metrics conflate. Applied to reduced GotenNet models on rMD17, it yields a clear empirical picture. The architecture preserves rotation, translation, and reflection to numerical precision (∼10−6 relative force error), so the auxiliary head leaves equivariance intact. The geometric targets, however, differ in parity: cosτ and |cosτ| are parity-even, while sinτ and the scalar triple product are parity-odd and near-zero-mean. The contrast between cosτ and |cosτ| reflects torsional-moment scaling, not parity. Across three-seed paired runs, the auxiliary effect on force mean absolute error (MAE) is dominated by seed noise. A seed-consistent molecule-dependent effect in raw energy MAE disappears when each model’s predicted energies are shifted by a constant, even though forces and conformer energy differences remain unchanged, showing how the bulk metric manufactures a gauge artifact. We offer preliminary design principles for symmetry-preserving auxiliary objectives rather than a new force-field benchmark.
Tensor-network methods have opened a powerful route for the study of dynamical spectral functions in two-dimensional quantum systems. However, existing approaches within the framework of infinite projected entangled-pair states construct the required renormalization tensors solely from the ground-state environment and can suffer from severe numerical instability. We identify the origin of this instability and introduce an excitation-tailored corner-transfer-matrix renormalization-group (ET-CTMRG) method to resolve it. By incorporating excitation tensors into the renormalization procedure, the method constructs a substantially more accurate effective Hamiltonian matrix and thereby yields reliable and well-converged excitation spectra. For Heisenberg antiferromagnets, it reduces truncation errors by orders of magnitude and for the particularly complex case of the supersolid phase in the triangular-lattice XXZ magnet $\mathrm{K_2Co(SeO_3)_2}$, it achieves excellent quantitative agreement with inelastic neutron-scattering measurements. ET-CTMRG therefore provides a robust framework for investigating the dynamical properties of strongly correlated quantum systems.
Jiahang Hu, Runze Chi, B. Normand et al.· 0 citations
Machine-learned interatomic potentials have enabled highly accurate atomistic simulations, but extending these capabilities to coarse-grained systems remains challenging due to the loss of geometric and orientational information during coarse-graining. In this work, we present a generalized framework for incorporating molecular geometry and directionality into coarse-grained machine-learned potentials through two complementary approaches: anisotropic density-based descriptors (AniSOAP) and symmetry-adapted equivariant message-passing neural networks (MACE-CG). Using Gay-Berne particles and coarse-grained representations of benzene, formamide, and water, we demonstrate that explicitly retaining molecular anisotropy substantially improves the prediction of energies, forces, and torques relative to isotropic representations. AniSOAP provides an effective linear baseline when molecular shape is well approximated by ellipsoidal symmetry, while symmetry-adapted MACE-CG enables the incorporation of arbitrary molecular point-group symmetries. For water, whose orientational degrees of freedom are poorly represented by ellipsoidal descriptors alone, symmetry-adapted rigid-body features improve energy, force, and torque prediction by resolving orientational degeneracies inherent to isotropic and moment-of-inertia-based representations. These results show that information loss in coarse-grained modeling is governed not only by mapping resolution but also by the symmetry and geometric information retained in the representation, providing a systematic route toward more expressive and transferable coarse-grained machine-learned potentials.
Arthur Y. Lin, Tejas Dahiya, Rose K. Cersonsky· 0 citations
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