Machine Learning Force-Field Approach for Itinerant Electron Magnets

TL;DR

This work develops a symmetry-informed machine-learning force-field framework for Landau-Lifshitz-Gilbert dynamics in itinerant magnets, focusing on descriptor constructions that respect global SO(3) spin rotation and lattice point-group symmetries. By learning local energies $\epsilon_i$ from neighborhood configurations and differentiating to obtain local fields $\mathbf H_i$, the authors enable scalable, accurate LLG simulations of complex spin textures in the s-d model on a triangular lattice. They introduce power-spectrum/bispectrum–based descriptors augmented by reference irreducible representations to manage symmetry and dimensionality, and validate the approach on 120°, tetrahedral, and triple-$Q$ skyrmion orders, including large-scale thermal quenches that reveal arrested ordering and glassy stripe states. The results demonstrate the practical utility of ML force-field methodologies for dynamical modeling of itinerant magnets and provide a pathway to apply these techniques to spin-orbit–coupled systems and more complex materials.

Abstract

We review the recent development of machine-learning (ML) force-field frameworks for Landau-Lifshitz-Gilbert (LLG) dynamics simulations of itinerant electron magnets, focusing on the general theory and implementations of symmetry-invariant representations of spin configurations. The crucial properties that such magnetic descriptors must satisfy are differentiability with respect to spin rotations and invariance to both lattice point-group symmetry and internal spin rotation symmetry. We propose an efficient implementation based on the concept of reference irreducible representations, modified from the group-theoretical power-spectrum and bispectrum methods. The ML framework is demonstrated using the s-d models, which are widely applied in spintronics research. We show that LLG simulations based on local fields predicted by the trained ML models successfully reproduce representative non-collinear spin structures, including 120$^\circ$, tetrahedral, and skyrmion crystal orders of the triangular-lattice s-d models. Large-scale thermal quench simulations enabled by ML models further reveal intriguing freezing dynamics and glassy stripe states consisting of skyrmions and bi-merons. Our work highlights the utility of ML force-field approach to dynamical modeling of complex spin orders in itinerant electron magnets.

Figures (8)

• Figure 1: Schematic diagram of ML force-field model for itinerant electron magnets. A descriptor transforms the neighborhood spin configuration $\mathcal{C}_i$ to effective coordinates $\{G_\ell \}$ which are then fed into a neural network (NN). The output node of the NN correspond to the local energy $\epsilon_i = \varepsilon(\mathcal{C}_i)$ associated with spin $\mathbf S_i$. The corresponding total potential energies $E$is obtained from summation of these local energies. Automatic differentiation is employed to compute the derivatives $\partial E / \partial \mathbf S_i$ from which the local exchange fields $\mathbf H_i$ are obtained.
• Figure 2: Symmetry-invariant descriptor for the neighborhood spin configuration. (a) The six bond variables $b_k = \mathbf S_0 \cdot \mathbf S_k$, constructed from the inner product of the six third-nearest neighboring spins $\mathbf S_k$ and the center spin $\mathbf S_0$, form the basis of a four-dimensional reducible representation of the D$_6$ point group. (b) Similarly, the six scalar chirality variables $\chi_1, \chi_2, \cdots ,\chi_6$, each indicated by a triangle, also form the basis of a reducible 6-dim representation of $D_6$. The scalar chirality of a triangle $(ijk)$ is $\chi = \mathbf S_i \cdot \mathbf S_j \times \mathbf S_k$. (c) An example of neighboring off-site bond variables forming a 12-dim reducible representation of the $D_6$ group. (d) Schematic diagram showing six symmetry-related blocks of neighboring sites within the cutoff radius $r_c$ of a center spin.
• Figure 3: Top row: Comparison between the component-wise ML-predicted torques $\mathbf T_{\rm ML} = \mathbf S \times \mathbf H_{\rm ML}$ and the ground truth $\mathbf T_{\rm exact}$ for s-d models that exhibit (a) the coplanar 120$^\circ$ order, (d) the non-coplanar tetrahedral order, and (g) a skyrmion crystal phase in the ground state. The blue and orange data points correspond to training and test datasets, respectively. The three panels (b), (e), (h) in the middle display the corresponding snapshots of spin configurations obtained from ML-based LLG simulations for these three cases. The bottom panels (c), (f), (i) show the ensemble-averaged spin structure factor $\mathcal{S}(\mathbf q)$ of these three phases from ML-LLG simulations.
• Figure 4: Snapshots of local scalar chirality $\chi_{ijk} = \mathbf S_i \cdot \mathbf S_j \times \mathbf S_k$, where $(ijk)$ denotes lattice sites on an elementary triangular unit, at various time steps after a thermal quench of the triangular s-d model at filling fraction close to 1/4.
• Figure 5: Comparison of time-dependent structure factors $\mathcal{S}(\mathbf q)$ obtained from KPM- and ML-based LLG simulations on a $48\times 48$ lattice. The structure factors at different times after a thermal quench are plotted along high symmetry directions $\Gamma(0,0)\rightarrow K(\frac{4\pi}{3},0)\rightarrow M(2\pi,0)$. The blue lines show values of $\mathcal{S}(\mathbf q)$ averaged over 50 independent KPM-LLG simulations, with the shaded area indicating the standard deviations. The red dots represent the average $\mathcal{S}(\mathbf q)$ from ML-LLG simulations with the corresponding standard deviation indicated by the error bars. Note the difference in $y$ scales at different times.