Long-Range Machine Learning of Electron Density for Twisted Bilayer Moir\'e Materials

Abstract

Moir\'e superlattices in two-dimensional (2D) materials exhibit rich quantum phenomena, but ab initio modelling of these systems remains computationally prohibitive. Existing machine learning methods for accelerating density-functional theory (DFT) can target the prediction of different quantities and often rely on the locality assumption. Here we train a Gaussian process regression SALTED model exclusively on the electron densities of small displaced bilayer structures and then extrapolate electron density prediction to the large supercells required to describe small twist angles between these bilayers. We show the necessity of long-range descriptors to yield reliable band structures and electrostatic properties of large twisted bilayer structures, when these are derived from predicted densities. We demonstrate that the choice of descriptor determines the distribution of residual density errors, which in turn affects the downstream electronic properties. We apply our models to twisted bilayer graphene, hexagonal boron nitride, and transition metal dichalcogenides, focusing on the model's capacity to predict complex phenomena, including flat band formation, bandwidth narrowing, domain-wall electric fields, and spin-orbit coupling effects. Beyond moir\'e materials, this approach provides a general methodology for electronic structure prediction in large-scale systems with substantial long-range phenomena related to non-local geometric information.