Active learning of effective Hamiltonian for super-large-scale atomic structures

TL;DR

This work introduces a general, first-principles-based effective Hamiltonian for perovskites and an on-the-fly active-learning framework that parameterizes it with Bayesian linear regression. The approach predicts energy, forces, and stress (and their uncertainties) during MD, triggering FP calculations to refine parameters when needed, enabling accurate simulations of giant systems with complex interactions. Demonstrations on BaTiO$_3$, Pb(Zr,Ti)O$_3$, and SrTiO$_3$/PbTiO$_3$ bilayers show improved phase diagrams, close agreement with experiments, and the discovery/validation of polar skyrmions in multilayer interfaces. The methodology significantly reduces FP workload and extends accurate, automated parameterization to super-large-scale atomic structures, supporting exploration of phase transitions and topological polar phenomena in complex multicomponent oxides.

Abstract

The first-principles-based effective Hamiltonian scheme provides one of the most accurate modeling technique for large-scale structures, especially for ferroelectrics. However, the parameterization of the effective Hamiltonian is complicated and can be difficult for some complex systems such as high-entropy perovskites. Here, we propose a general form of effective Hamiltonian and develop an active machine learning approach to parameterize the effective Hamiltonian based on Bayesian linear regression. The parameterization is employed in molecular dynamics simulations with the prediction of energy, forces, stress and their uncertainties at each step, which decides whether first-principles calculations are executed to retrain the parameters. Structures of BaTiO$_3$, Pb(Zr$_{0.75}$Ti$_{0.25}$)O$_3$ and (Pb,Sr)TiO$_3$ system are taken as examples to show the accuracy of this approach, as compared with conventional parametrization method and experiments. This machine learning approach provides a universal and automatic way to compute the effective Hamiltonian parameters for any considered complex systems with super-large-scale (more than $10^7$ atoms) atomic structures.

Figures (4)

• Figure 1: Schematics of on-the-fly learning of effective Hamiltonian for perovskite structure. (a) The projection between the atomic structure used in the FP calculations and the order-parameters configuration used in effective Hamiltonian MD simulations. (b) The workflow of on-the-fly learning of effective Hamiltonian.
• Figure 2: On-the-fly machine learning of parametrization for BaTiO3. (a, b, c) (a) Bayesian error, (b) potential energy per formal unit (f.u.), and (c) local dipolar mode $\bm u$ in the learning/fitting process as functions of MD steps. The dash line in panel (a) denotes the threshold to perform FP calculations. The blue and orange lines in panel (b) represent the energy computed by effective Hamiltonian model during the fitting process and with the the final parameter after the learning process, respectively. Phase diagram by effective Hamiltonian simulations with the parameters from (d) FP calculations in Ref. BTO_Zhong1995 and (e) on-the-fly learning. Absolute values of local mode $\bm u$ of BaTiO3 as functions of temperature are shown in panel (d) and (e). Here, R, O, T, and C denote the rhombohedral, orthogonal, tetrahedral and cubic phases, respectively.
• Figure 3: Polar distribution of SrTiO3/PbTiO3 bilayer. (a) Dipole configuration of SrTiO3/PbTiO3 averaged over the top 10 planes of PbTiO3 layer obtained from HMC simulation with the effective Hamiltonian. The arrows denote the in-plane component of the local dipolar mode, and the colors denote the out-of-plane components of dipolar modes. (b) The dipole configuration of SrTiO3/PbTiO3 in a (010) plane around the circled skyrmion in panel (a). The colors of the arrows denote the out-of-plane component of local dipolar mode. (c) The schematic of the skyrmion at the top of the PbTiO3 layer, where the color of the arrows denote the out-of-plane component of local dipolar mode. (d-h) The PFM images characterizing the skyrmion in SrTiO3/PbTiO3. (d) Vertical PFM amplitude and phase. (e-h) Lateral PFM amplitudes and phases corresponding to the range marked with red rectangles in panel (c), with tip orientation angles of 0° (e), 30° (f), 60° (g) and 90° (h). The white circles in panels (d-h) mark one of the skyrmion. The arrows in the circles in panels (e-h) mark the in-plane component of the polarization.
• Figure 4: Computational time for 100 MD steps calculations as a function of the number of atoms in the simulated of BaTiO3 supercell, using the effective Hamiltonian ($H_{\text{eff}}$), MLFF, eep potential MD,and ab-initio MD (AIMD). The tests are performed on Intel(R) Xeon(R) Silver 4210R CPU using one core, expect for the AIMD simulation, which is performed on the Intel(R) Xeon(R) CPU E5-2680 v3 CPU using 24 cores.