Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems

Gia-Wei Chern; Yunhao Fan; Sheng Zhang; Puhan Zhang

cond-mat.str-elApr 13, 2026

Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems

Authors: Gia-Wei Chern, Yunhao Fan, Sheng Zhang, Puhan Zhang

Organizations: Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Abstract

We review recent advances in machine-learning (ML) force-field methods for large-scale Landau-Lifshitz-Gilbert (LLG) simulations of metallic spin systems. We generalize the Behler-Parrinello (BP) ML architecture -- originally developed for quantum molecular dynamics -- to construct scalable and transferable ML models capable of capturing the intricate dependence of electron-mediated exchange fields on the local magnetic environment characteristic of itinerant magnets. A central ingredient of this framework is the implementation of symmetry-aware magnetic descriptors based on group-theoretical bispectrum formalisms. Leveraging these ML force fields, LLG simulations faithfully reproduce hallmark non-collinear magnetic orders -- such as the $120^\circ$ and tetrahedral states -- on the triangular lattice, and successfully capture the complex spin textures emerging in the mixed-phase states of a square-lattice double-exchange model under thermal quench. We further discuss a generalized potential theory that extends the BP formalism to incorporate both conservative and nonconservative electronic torques, thereby enabling ML models to learn nonequilibrium exchange fields from computationally demanding microscopic approaches such as nonequilibrium Green's-function techniques. This extension yields quantitatively accurate predictions of voltage-driven domain-wall motion and establishes a foundation for quantum-accurate, multiscale modeling of nonequilibrium spin dynamics and spintronic functionalities.

Figures & tables

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 $\mathbf{S}_{i}$ . $\{x^{(1)}_{m}\}$ and $\{x^{(2)}_{n}\}$ denote feature variables associated with neurons of the first and second hidden layers, respectively. 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: (a) The four bond variables $b_{A},b_{B},b_{C},b_{D}$ , defined as the inner products between the central spin $\mathbf{S}_{i}$ and its four equidistant nearest neighbors, form the basis of a four-dimensional reducible representation of the $D_{4}$ point group. (b) The four scalar chiralities $\chi_{A},\chi_{B},\chi_{C},\chi_{D}$ likewise span a four-dimensional reducible representation of $D_{4}$ . (c) The eight bond variables $b_{A},b_{B},\ldots,b_{H}$ constitute the basis of an eight-dimensional reducible representation.

Figure 3: Top panels: (a) The non-collinear 120 ∘ magnetic order with the tripled unit cell of the half-filled triangular-lattice s–d model. (b) Benchmark of ML-predicted torques, $\mathbf{T}_{\rm ML}=\mathbf{S}\times\mathbf{H}_{\rm ML}$ , evaluated for the 120 ∘ state. (c) Ensemble-averaged spin structure factor $\mathcal{S}(\mathbf{q})$ obtained from ML-LLG simulations. Bottom panels: Results for the non-coplanar tetrahedral order at filling fraction $f\approx 1/4$ : (d) real-space spin configuration with a quadrupled magnetic unit cell; (e) ML torque-prediction benchmark; and (f) the corresponding ensemble-averaged spin structure factor.

Figure 4: (a) ML-predicted exchange fields compared against exact values from the test dataset. (b) Distribution of the force difference $\delta=H_{\text{ML}}-H_{\text{exact}}$ , which is well described by a normal distribution (red curve) with variance $\sigma^{2}=0.035$ .

Figure 5: (Top) Density plots of the local nearest-neighbor spin correlation $b_{i}\equiv(\mathbf{S}_{i}\cdot\mathbf{S}_{i+\mathbf{x}}+\mathbf{S}_{i}\cdot\mathbf{S}_{i-\mathbf{x}}+\mathbf{S}_{i}\cdot\mathbf{S}_{i+\mathbf{y}}+\mathbf{S}_{i}\cdot\mathbf{S}_{i-\mathbf{y}})/4$ at four representative times during the ML-LLG simulation on a $200\times 200$ lattice. (Bottom) Corresponding ML-predicted distributions of the local electron number $n_{i}=\sum_{\alpha}\langle\hat{c}^{\dagger}_{i\alpha}\hat{c}^{\,}_{i\alpha}\rangle$ . The neural-network model is trained on datasets obtained from exact solutions for an $L=30$ lattice with parameters $t{\rm nn}=1$ , $J=7$ , and electron filling fraction $n=0.485$ .

Figure 6: Average linear size

L=\langle s\rangle^{1/2}

of ferromagnetic (FM) clusters as a function of time following a thermal quench. The dash–dot line marks the power-law growth

L(t)\sim t^{1/3}

, while the dashed line shows a sublogarithmic form

L(t)\sim(\log t)^{\beta}

with

\beta=0.11

. The inset displays the corresponding time evolution of the average cluster size

\langle s\rangle

Figure 7: A tangential vector field on a sphere can be decomposed into (a) the curl-free component $\nabla_{s}\,{E}$ and (b) the divergence-free component $\nabla_{s}\times{G}$ . (c) shows the gradient field $\mathbf{h}^{\rm eq}_{i}=-\partial{E}/\partial\mathbf{S}_{i}$ , which can be viewed as a quasi-equilibrium exchange field, and the curl-field $\mathbf{h}^{\rm neq}_{i}=-\mathbf{S}_{i}\times\partial{G}/\partial\mathbf{S}_{i}$ , which corresponds to the nonequilibrium force, and their respective torques $\mathbf{T}^{\rm eq}_{i}=\mathbf{h}^{\rm eq}_{i}\times\mathbf{S}_{i}$ and $\mathbf{T}^{\rm neq}_{i}=\mathbf{h}^{\rm neq}_{i}\times\mathbf{S}_{i}$ .

Figure 8: (a) Schematic diagram of neural-network (NN) force-field model for out-of-equilibrium itinerant spin system. A descriptor transforms the neighborhood spin configuration $\mathcal{C}_{i}$ to effective coordinates $\{G_{\ell}\}$ which are then fed into a NN. The two output nodes of the NN correspond to the local energy $\epsilon_{i}=\varepsilon(\mathcal{C}_{i})$ and $\gamma_{i}=\chi(\mathcal{C}_{i})$ associated with site- $i$ . The corresponding total potential energies ${E}$ and ${G}$ are obtained from summation of these local energies. Automatic differentiation is employed to compute the derivatives $\partial{E}/\partial\mathbf{S}_{i}$ and $\partial{G}/\partial\mathbf{S}_{i}$ , from which the local exchange fields $\mathbf{H}_{i}$ are obtained according to the generalized force expression Eq.( 21 ).

Figure 9: (Color online) Schematic diagram of the voltage-driven magnetic transition in the DE model sandwiched by two electrodes. In the initial state, spins in the DE model are arranged in a staggered Néel order. A voltage drop $V$ is applied to the two electrodes.

Figure 10: (a) The ML predicted forces versus the exact solution from the NEGF calculation for the s-d model with exchange coupling $J=3.8t_{\rm nn}$ . (b) Normalized distribution of the prediction error of the perpendicular components of the forces.

Figure 11: Domain wall propagation in a s-d system driven by an external voltage $eV=3.2t_{\rm nn}$ . Comparison between (a) NEGF-LLG simulations and (b) ML-LLG simulations. The lattice size is $30\times 24$ . The color bar indicates the local nearest-neighbor spin correlation $b_{ij}=\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ . Blue (red) area corresponds to AFM (FM) domains. The NEGF-LLG simulation was carried out at a low temperature of $k_{B}T=0.01\,t_{\rm nn}$ , while the ML simulation was performed without Langevin noise.

Figure 12: (a) Average position of FM-AFM domain, obtained from NEGF-LLG and ML-LLG simulations, as a function of time during the voltage-driven insulator to metal transition of the s-d model. (b) histogram of the ratio $|\mathbf{T}^{\rm neq}_{i}|/|\mathbf{T}^{\rm eq}_{i}|$ predicted by the NN model for the simulation of domain-wall propagation. Here $\mathbf{T}^{\rm eq}_{i}=\mathbf{S}_{i}\times\partial{E}/\partial\mathbf{S}_{i}$ is the quasi-equilibrium torque, and $\mathbf{T}^{\rm neq}_{i}=\mathbf{S}_{i}\times(\mathbf{S}_{i}\times\partial{G}/\partial\mathbf{S}_{i})$ is the non-equilibrium torque in the generalized LLG Eq. ( VI.1 ).

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

Gia-Wei Chern, Yunhao Fan, Sheng Zhang, Puhan Zhang

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Materials Science

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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.

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