Coarsening of chiral domains in itinerant electron magnets: A machine learning force field approach

TL;DR

This work tackles the coarsening of chiral domains in itinerant magnets on a triangular lattice by developing a scalable machine-learning force field within the Behler-Parrinello framework. The model learns the local electron-driven energy $E=\sum_i \epsilon_i$ and yields accurate local fields $\mathbf H_i$ to drive large-scale LLG dynamics, validated against kernel polynomial method benchmarks. Applying this to the Kondo-lattice model, the authors uncover a nearly linear growth of chiral domains at late times, driven by anisotropic domain walls and the nonconserved nature of the chiral order parameter, with dynamical scaling and Porod-like behavior observed in structure factors. The approach enables efficient, quantitative exploration of phase ordering in itinerant magnets and points to improved macroscopic phase-field models that account for directional interfaces and vertex effects.

Abstract

Frustrated itinerant magnets often exhibit complex noncollinear or noncoplanar magnetic orders which support topological electronic structures. A canonical example is the anomalous quantum Hall state with a chiral spin order stabilized by electron-spin interactions on a triangular lattice. While a long-range magnetic order cannot survive thermal fluctuations in two dimensions, the chiral order which results from the breaking of a discrete Ising symmetry persists even at finite temperatures. We present a scalable machine learning (ML) framework to model the complex electron-mediated spin-spin interactions that stabilize the chiral magnetic domains in a triangular lattice. Large-scale dynamical simulations, enabled by the ML force-field models, are performed to investigate the coarsening of chiral domains after a thermal quench. While the chiral phase is described by a broken $Z_2$ Ising-type symmetry, we find that the characteristic size of chiral domains increases linearly with time, in stark contrast to the expected Allen-Cahn domain growth law for a non-conserved Ising order parameter field. The linear growth of the chiral domains is attributed to the orientational anisotropy of domain boundaries. Our work also demonstrates the promising potential of ML models for large-scale spin dynamics of itinerant 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 $\{ p^\Gamma_r, \eta^\Gamma_r \}$ 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: Benchmark of ML for the adiabatic dynamics of the KLM. Panel (a) compares components of the ML-predicted torque $T_{\rm ML}$ and the ground truth $T_{\rm ED}$ obtained from exact diagonalization (ED). The torque is defined as $\mathbf T_i = \mathbf S_i \times \mathbf H_i$, where $\mathbf H_i$ is the local effective field for the $i$-th spin. Panel (b) shows the histograms of the prediction error $\delta =T_{\rm ED} - T_{\rm ML}.$
• Figure 3: Chirality correlation functions $C_{\chi}(\mathbf r)$ at various times after a thermal quench at $t = 0$. These results are obtained from LLG simulations on a $96\times 96$ lattice. The red solid and blue open circles are data points with local fields computed using the ML force-field model and the KPM, respectively.
• Figure 4: Snapshots of local chirality parameter $\chi_{\triangle}(\mathbf r)$ at various time steps after a thermal quench of the triangular KLM at $n\sim 1/4$ filling. The color intensity indicates the local chirality normalized by its amplitude $\chi_{\rm tet} = 4 S^3 / 3 \sqrt{3}$ of the tetrahedral spin structure. An initially random configuration is suddenly quenched to a zero temperature $T = 0$ at time $t = 0$. The ML-LLG dynamics is used to simulate the relaxation of the system toward equilibrium.
• Figure 5: Histogram of local spin chirality (in units of $S^3$) defined on triangular plaquettes at various times after a thermal quench. The late stage of phase ordering is characterized by a pronounced bimodal distribution with two strong peaks at $\pm \chi_{\rm tet}$, where $\chi_{\rm tet} = 4 S^3 / 3 \sqrt{3} \approx 0.77 S^3$ is the chirality of the tetrahedral spin order.