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.

Machine learning predictions for local electronic properties of disordered correlated electron systems

We present a scalable machine learning (ML) model to predict local electronic properties such as on-site electron number and double occupation for disordered correlated electron systems. Our approach is based on the locality principle, or the nearsightedness nature, of many-electron systems, which means local electronic properties depend mainly on the immediate environment. A ML model is developed to encode this complex dependence of local quantities on the neighborhood. We demonstrate our approach using the square-lattice Anderson-Hubbard model, which is a paradigmatic system for studying the interplay between Mott transition and Anderson localization. We develop a lattice descriptor based on group-theoretical method to represent the on-site random potentials within a finite region. The resultant feature variables are used as input to a multi-layer fully connected neural network, which is trained from datasets of variational Monte Carlo (VMC) simulations on small systems. We show that the ML predictions agree reasonably well with the VMC data. Our work underscores the promising potential of ML methods for multi-scale modeling of correlated electron systems.

Descriptors for Machine Learning Model of Generalized Force Field in Condensed Matter Systems

We outline the general framework of machine learning (ML) methods for multi-scale dynamical modeling of condensed matter systems, and in particular of strongly correlated electron models. Complex spatial temporal behaviors in these systems often arise from the interplay between quasi-particles and the emergent dynamical classical degrees of freedom, such as local lattice distortions, spins, and order-parameters. Central to the proposed framework is the ML energy model that, by successfully emulating the time-consuming electronic structure calculation, can accurately predict a local energy based on the classical field in the intermediate neighborhood. In order to properly include the symmetry of the electron Hamiltonian, a crucial component of the ML energy model is the descriptor that transforms the neighborhood configuration into invariant feature variables, which are input to the learning model. A general theory of the descriptor for the classical fields is formulated, and two types of models are distinguished depending on the presence or absence of an internal symmetry for the classical field. Several specific approaches to the descriptor of the classical fields are presented. Our focus is on the group-theoretical method that offers a systematic and rigorous approach to compute invariants based on the bispectrum coefficients. We propose an efficient implementation of the bispectrum method based on the concept of reference irreducible representations. Finally, the implementations of the various descriptors are demonstrated on well-known electronic lattice models.

Machine learning nonequilibrium electron forces for adiabatic spin dynamics

We present a generalized potential theory of nonequilibrium torques for the Landau-Lifshitz equation. The general formulation of exchange forces in terms of two potential energies allows for the implementation of accurate machine learning models for adiabatic spin dynamics of out-of-equilibrium itinerant magnetic systems. To demonstrate our approach, we develop a deep-learning neural network that successfully learns the forces in a driven s-d model computed from the nonequilibrium Green's function method. We show that the Landau-Lifshitz dynamics simulations with forces predicted from the neural-net model accurately reproduce the voltage-driven domain-wall propagation. Our work opens a new avenue for multi-scale modeling of nonequilibrium dynamical phenomena in itinerant magnets and spintronics based on machine-learning models.

Anomalous phase separation and hidden coarsening of super-clusters in the Falicov-Kimball model

We show that the celebrated Falicov-Kimball model exhibits rich and intriguing phase-ordering dynamics. Applying modern machine learning methods to enable large-scale quantum kinetic Monte Carlo simulations, we uncover an unusual phase-separation scenario in which the growth of charge checkerboard clusters competes with domain coarsening related to a hidden symmetry-breaking. A self-trapping mechanism as a result of this competition gives rise to arrested growth of checkerboard patterns and their super-clusters. Glassy behaviors similar to the one reported in this work could be generic for other correlated electron systems.

Arrested phase separation in double-exchange models: machine-learning enabled large-scale simulation

We present large-scale dynamical simulations of electronic phase separation in the single-band double-exchange model based on deep-learning neural-network potentials trained from small-size exact diagonalization solutions. We uncover an intriguing correlation-induced freezing behavior as doped holes are segregated from half-filled insulating background during equilibration. While the aggregation of holes is stabilized by the formation of ferromagnetic clusters through Hund's coupling between charge carriers and local magnetic moments, this stabilization also creates confining potentials for holes when antiferromagnetic spin-spin correlation is well developed in the background. The dramatically reduced mobility of the self-trapped holes prematurely disrupts further growth of the ferromagnetic clusters, leading to an arrested phase separation. Implications of our findings for phase separation dynamics in materials that exhibit colossal magnetoresistance effect are discussed.

Machine learning dynamics of phase separation in correlated electron magnets

We demonstrate machine-learning enabled large-scale dynamical simulations of electronic phase separation in double-exchange system. This model, also known as the ferromagnetic Kondo lattice model, is believed to be relevant for the colossal magnetoresistance phenomenon. Real-space simulations of such inhomogeneous states with exchange forces computed from the electron Hamiltonian can be prohibitively expensive for large systems. Here we show that linear-scaling exchange field computation can be achieved using neural networks trained by datasets from exact calculation on small lattices. Our Landau-Lifshitz dynamics simulations based on machine-learning potentials nicely reproduce not only the nonequilibrium relaxation process, but also correlation functions that agree quantitatively with exact simulations. Our work paves the way for large-scale dynamical simulations of correlated electron systems using machine-learning models.