Machine-learning force-field models for dynamical simulations of metallic magnets

TL;DR

These results establish ML force-field frameworks as scalable, accurate, and versatile tools for modeling nonequilibrium spin dynamics in itinerant magnets.

Abstract

We review recent advances in machine learning (ML) force-field methods for Landau-Lifshitz-Gilbert (LLG) simulations of itinerant electron magnets, focusing on scalability and transferability. Built on the principle of locality, a deep neural network model is developed to efficiently and accurately predict the electron-mediated forces governing spin dynamics. Symmetry-aware descriptors constructed through a group-theoretical approach ensure rigorous incorporation of both lattice and spin-rotation symmetries. The framework is demonstrated using the prototypical s-d exchange model widely employed in spintronics. ML-enabled large-scale simulations reveal novel nonequilibrium phenomena, including anomalous coarsening of tetrahedral spin order on the triangular lattice and the freezing of phase separation dynamics in lightly hole-doped, strong-coupling square-lattice systems. These results establish ML force-field frameworks as scalable, accurate, and versatile tools for modeling nonequilibrium spin dynamics in itinerant magnets.

Figures (5)

• 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 $\{ p^\Gamma_r, \eta^\Gamma_r \}$ which are then fed into a neural network (NN). The output node of the NN corresponds to the local energy $\epsilon_i = \varepsilon(\mathcal{C}_i)$ associated with spin $\mathbf S_i$. The corresponding total potential energy $E$ is obtained from the 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: (a) Tetrahedral spin order on a triangular lattice [Eq. (\ref{['eq:triple-Q']})]. (b) Two inequivalent spins in a quadrupled unit cell with opposite chirality. (c) Comparison of ML-predicted torque $T_{\rm ML}$ with KPM results $T_{\rm KPM}$, where $\mathbf T_i = \mathbf S_i \times \mathbf H_i$. (d) Histogram of prediction error $\delta = T_{\rm KPM} - T_{\rm ML}$.
• Figure 3: (a) Snapshots of local scalar chirality $\chi_{ijk} = \mathbf S_i \cdot \mathbf S_j \times \mathbf S_k$, with $(ijk)$ denoting sites of an elementary triangle, at different times after a thermal quench of the triangular s–d model near $n=1/4$. (b) Characteristic length $L$ versus time, extracted from the chirality structure factor (linear scales on both axes). Yellow triangles and green circles show data from LLG quenches using KPM and ML-calculated fields, respectively. The ML model was trained only on KPM–LLG data up to $t_{\rm training} = 800$, marked by the yellow shaded region.
• Figure 4: (a) ML-predicted exchange fields versus exact results from the test dataset. (b) The distribution of the force difference $\delta = H_{\text{ML}} - H_{\text{exact}}$, well fitted by a normal distribution (red line) with variance $\sigma^2 = 0.035$. (c) Spin-spin correlation $\langle \mathbf S_i \cdot \mathbf S_j \rangle$ as a function of distance $r_{ij} = |\mathbf r_j - \mathbf r_i|$ along the $x$ direction at filling $f = 0.485$. Red dots show LLG results using NN models without Langevin noise, while blue lines correspond to ED-LLG simulations at $T = 0.022$.
• Figure 5: (a) Density maps of local bond variables $b(\mathbf r_i)$ at four different times during the ML–LLG simulation on a $100\times100$ lattice with $1.5\%$ hole doping. (b) Average linear size $L = \langle s \rangle^{1/2}$ of FM clusters versus time after a thermal quench. The dash-dotted line denotes the $t^{1/3}$ power-law growth, while the dashed line shows a sublogarithmic dependence $L(t) \sim (\log t)^{\beta}$ with $\beta = 0.11$. The inset plots the time evolution of the average cluster size $\langle s \rangle$.