Machine learning force-field model for kinetic Monte Carlo simulations of itinerant Ising magnets

TL;DR

This work introduces a locality-driven machine-learning framework to enable large-scale kinetic Monte Carlo simulations of itinerant Ising magnets by predicting the local field $h_i$ and energy change $\Delta E_i = 2 \sigma_i h_i$ from a finite neighborhood using a convolutional neural network. The fixed-size CNN ensures $O(N)$ computational cost per Monte Carlo sweep, allowing simulations on lattices far beyond reach of exact electronic-structure methods. Benchmarks against exact diagonalization show the ML model accurately reproduces both equilibrium thermodynamics (e.g., $T_c$) and dynamical evolution, including correlation functions after quenches. Large-scale coarsening dynamics reveal temperature-dependent domain growth exponents, including anomalous $\alpha=1/4$ behavior at low temperatures, and demonstrate dynamical scaling despite long-range, itinerant-electron-mediated interactions. The approach offers a transferable, scalable path for modeling discrete dynamical systems coupled to quantum degrees of freedom in multi-scale contexts.

Abstract

We present a scalable machine learning (ML) framework for large-scale kinetic Monte Carlo (kMC) simulations of itinerant electron Ising systems. As the effective interactions between Ising spins in such itinerant magnets are mediated by conducting electrons, the calculation of energy change due to a local spin update requires solving an electronic structure problem. Such repeated electronic structure calculations could be overwhelmingly prohibitive for large systems. Assuming the locality principle, a convolutional neural network (CNN) model is developed to directly predict the effective local field and the corresponding energy change associated with a given spin update based on Ising configuration in a finite neighborhood. As the kernel size of the CNN is fixed at a constant, the model can be directly scalable to kMC simulations of large lattices. Our approach is reminiscent of the ML force-field models widely used in first-principles molecular dynamics simulations. Applying our ML framework to a square-lattice double-exchange Ising model, we uncover unusual coarsening of ferromagnetic domains at low temperatures. Our work highlights the potential of ML methods for large-scale modeling of similar itinerant systems with discrete dynamical variables.

Figures (7)

• Figure 1: Schematic diagram of ML model for prediction of a local effective field $h_i$ associated with a spin $\sigma_i$ on the lattice. The ML model consists of a convolutional neural network (CNN) and a fully connected network. The input to the CNN is Ising configurations $\sigma_j$ within a finite block $\mathcal{B}_i$ in the neighborhood of site-$i$. The single node at the output layer of the fully connected neural net gives the local field $h_i$. The energy change caused by the local spin flip $\sigma_i \to -\sigma_i$ is given by $\Delta E_i = 2 \sigma_i h_i$.
• Figure 2: Benchmark of the ML prediction for the energy difference $\Delta E$. Panel (a) compares the ML-predicted energy difference $\Delta E_{\rm ML}$ to the ground-truth value $\Delta E_{\rm ED}$ calculated by exact diagonalization. The mean squared error on the test set is 0.0014. Panel (b) shows a histogram of the prediction error $\delta = \Delta E_{\rm ML} - \Delta E_{\rm ED}$ on the test dataset.
• Figure 3: MCMC simulation results based on the ML energy model for the Ising-DE system. (a) magnetization $M$ and (b) magnetic susceptibility $\chi$ versus temperature. The system size is $30 \times 30$ and the temperature is measured in unit of the nearest-neighbor hopping $t_{\rm nn}$.
• Figure 4: Dynamical benchmark comparison of ML and ED correlation functions at various times after a quench to temperature $T = 0.01$ for a $30 \times 30$ lattice. $n_{step}$ refers to the number of kinetic Monte Carlo spin-update attempts performed before calculating the correlation function. Correlation functions are averaged over 30 independent runs and error bars of $\pm 1$ standard deviation are shown.
• Figure 5: Snapshots of local magnetization $m$ at various times $t$ after a thermal quench of a $200 \times 200$ spin system to temperatures $T=0.1$ and $T=0.01$. The red (blue) regions correspond to ferromagnetic domains of $\sigma_i = +1$ ($\sigma_i = -1$) spins. The system began in a random configuration, and kinetic Monte Carlo simulations with Glauber spin-flip dynamics were paired with the ML $\Delta E$ predictions to simulate its evolution after a sudden temperature quench. $t$ is defined as $n_{\rm step} / N$, where $n_{\rm step}$ is the number of kinetic Monte Carlo spin-update attempts and $N$ is the number of spins in the system.