Machine Learning Force-Field Approach for Itinerant Electron Magnets

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.

Enhanced coarsening of charge density waves induced by electron correlation: Machine-learning enabled large-scale dynamical simulations

The phase ordering kinetics of emergent orders in correlated electron systems is a fundamental topic in non-equilibrium physics, yet it remains largely unexplored. The intricate interplay between quasiparticles and emergent order-parameter fields could lead to unusual coarsening dynamics that is beyond the standard theories. However, accurate treatment of both quasiparticles and collective degrees of freedom is a multi-scale challenge in dynamical simulations of correlated electrons. Here we leverage modern machine learning (ML) methods to achieve a linear-scaling algorithm for simulating the coarsening of charge density waves (CDWs), one of the fundamental symmetry breaking phases in functional electron materials. We demonstrate our approach on the square-lattice Hubbard-Holstein model and uncover an intriguing enhancement of CDW coarsening which is related to the screening of on-site potential by electron-electron interactions. Our study provides fresh insights into the role of electron correlations in non-equilibrium dynamics and underscores the promise of ML force-field approaches for advancing multi-scale dynamical modeling of correlated electron systems.

Echo State network for coarsening dynamics of charge density waves

An echo state network (ESN) is a type of reservoir computer that uses a recurrent neural network with a sparsely connected hidden layer. Compared with other recurrent neural networks, one great advantage of ESN is the simplicity of its training process. Yet, despite the seemingly restricted learnable parameters, ESN has been shown to successfully capture the spatial-temporal dynamics of complex patterns. Here we build an ESN to model the coarsening dynamics of charge-density waves (CDW) in a semi-classical Holstein model, which exhibits a checkerboard electron density modulation at half-filling stabilized by a commensurate lattice distortion. The inputs to the ESN are local CDW order-parameters in a finite neighborhood centered around a given site, while the output is the predicted CDW order of the center site at the next time step. Special care is taken in the design of couplings between hidden layer and input nodes to ensure lattice symmetries are properly incorporated into the ESN model. Since the model predictions depend only on CDW configurations of a finite domain, the ESN is scalable and transferrable in the sense that a model trained on dataset from a small system can be directly applied to dynamical simulations on larger lattices. Our work opens a new avenue for efficient dynamical modeling of pattern formations in functional electron materials.

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

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.

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

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.