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.

Machine learning approach for vibronically renormalized electronic band structures

We present a machine learning (ML) method for efficient computation of vibrational thermal expectation values of physical properties from first principles. Our approach is based on the non-perturbative frozen phonon formulation in which stochastic Monte Carlo algorithm is employed to sample configurations of nuclei in a supercell at finite temperatures based on a first-principles phonon model. A deep-learning neural network is trained to accurately predict physical properties associated with sampled phonon configurations, thus bypassing the time-consuming {\em ab initio} calculations. To incorporate the point-group symmetry of the electronic system into the ML model, group-theoretical methods are used to develop a symmetry-invariant descriptor for phonon configurations in the supercell. We apply our ML approach to compute the temperature dependent electronic energy gap of silicon based on density functional theory (DFT). We show that, with less than a hundred DFT calculations for training the neural network model, an order of magnitude larger number of sampling can be achieved for the computation of the vibrational thermal expectation values. Our work highlights the promising potential of ML techniques for finite temperature first-principles electronic structure methods.

Kinetics of orbital ordering in cooperative Jahn-Teller models: Machine-learning enabled large-scale simulations

We present a scalable machine learning (ML) force-field model for the adiabatic dynamics of cooperative Jahn-Teller (JT) systems. Large scale dynamical simulations of the JT model also shed light on the orbital ordering dynamics in colossal magnetoresistance manganites. The JT effect in these materials describes the distortion of local oxygen octahedra driven by a coupling to the orbital degrees of freedom of $e_g$ electrons. An effective electron-mediated interaction between the local JT modes leads to a structural transition and the emergence of long-range orbital order at low temperatures. Assuming the principle of locality, a deep-learning neural-network model is developed to accurately and efficiently predict the electron-induced forces that drive the dynamical evolution of JT phonons. A group-theoretical method is utilized to develop a descriptor that incorporates the combined orbital and lattice symmetry into the ML model. Large-scale Langevin dynamics simulations, enabled by the ML force-field models, are performed to investigate the coarsening dynamics of the composite JT distortion and orbital order after a thermal quench. The late-stage coarsening of orbital domains exhibits pronounced freezing behaviors which are likely related to the unusual morphology of the domain structures. Our work highlights a promising avenue for multi-scale dynamical modeling of correlated electron systems.

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.

Machine learning force-field models for metallic spin glass

Metallic spin glass systems, such as dilute magnetic alloys, are characterized by randomly distributed local moments coupled to each other through a long-range electron-mediated effective interaction. We present a scalable machine learning (ML) framework for dynamical simulations of metallic spin glasses. A Behler-Parrinello type neural-network model, based on the principle of locality, is developed to accurately and efficiently predict electron-induced local magnetic fields that drive the spin dynamics. A crucial component of the ML model is a proper symmetry-invariant representation of local magnetic environment which is direct input to the neural net. We develop such a magnetic descriptor by incorporating the spin degrees of freedom into the atom-centered symmetry function methods which are widely used in ML force-field models for quantum molecular dynamics. We apply our approach to study the relaxation dynamics of an amorphous generalization of the s-d model. Our work highlights the promising potential of ML models for large-scale dynamical modeling of itinerant magnets with quenched disorder.

Machine learning for structure-property relationships: Scalability and limitations

We present a scalable machine learning (ML) framework for predicting intensive properties and particularly classifying phases of many-body systems. Scalability and transferability are central to the unprecedented computational efficiency of ML methods. In general, linear-scaling computation can be achieved through the divide and conquer approach, and the locality of physical properties is key to partitioning the system into sub-domains that can be solved separately. Based on the locality assumption, ML model is developed for the prediction of intensive properties of a finite-size block. Predictions of large-scale systems can then be obtained by averaging results of the ML model from randomly sampled blocks of the system. We show that the applicability of this approach depends on whether the block-size of the ML model is greater than the characteristic length scale of the system. In particular, in the case of phase identification across a critical point, the accuracy of the ML prediction is limited by the diverging correlation length. The two-dimensional Ising model is used to demonstrate the proposed framework. We obtain an intriguing scaling relation between the prediction accuracy and the ratio of ML block size over the spin-spin correlation length. Implications for practical applications are also discussed.

Machine learning for phase ordering dynamics of charge density waves

We present a machine learning (ML) framework for large-scale dynamical simulations of charge density wave (CDW) states. The charge modulation in a CDW state is often accompanied by a concomitant structural distortion, and the adiabatic evolution of a CDW order is governed by the dynamics of the lattice distortion. Calculation of the electronic contribution to the driving forces, however, is computationally very expensive for large systems. Assuming the principle of locality for electron systems, a neural-network model is developed to accurately and efficiently predict local electronic forces with input from neighborhood configurations. Importantly, the ML model makes possible a linear complexity algorithm for dynamical simulations of CDWs. As a demonstration, we apply our approach to investigate the phase ordering dynamics of the Holstein model, a canonical system of CDW order. Our large-scale simulations uncover an intriguing growth of the CDW domains that deviates significantly from the expected Allen-Cahn law for phase ordering of Ising-type order parameter field. This anomalous domain-growth could be attributed to the complex structure of domain-walls in this system. Our work highlights the promising potential of ML-based force-field models for dynamical simulations of functional electronic materials.