AI Predicts Twisted Material Behavior

Scientists are tackling the computational challenges of modelling complex phenomena in twisted bilayer Moiré materials, where traditional methods prove excessively demanding. Zekun Lou from MPI for the Structure and Dynamics of Matter, Alan M. Lewis from the Department of Chemistry at the University of York, and Mariana Rossi from both MPI for the Structure and Dynamics of Matter and the Yusuf Hamied Department of Chemistry, alongside et al., present a novel machine learning approach to predict electron densities in these systems. Their research significantly advances the field by moving beyond locality assumptions and employing long-range descriptors, enabling accurate predictions of crucial properties like band structure and electrostatic potential in large supercells. This methodology not only facilitates the study of exotic behaviours in materials such as graphene and hexagonal boron nitride, including flat band formation and domain-wall electric fields, but also establishes a general framework for electronic structure prediction in large-scale systems governed by non-local geometric information.

Scientists often rely on quantities and the locality assumption. We trained a Gaussian process regression SALTED model exclusively on the electron densities of small displaced bilayer structures and then extrapolated 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 the choice.

Descriptor selection and residual error distribution in moiré material modelling

Scientists have demonstrated that the choice of descriptor determines the distribution of residual density errors, which in turn affects the downstream electronic properties. They applied their 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 moiré 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. The electronic structure of two-dimensional (2D) materials exhibits significant quantum confinement effects, making them different from their bulk counterparts.

When two or more monolayers are stacked with a small twist angle, large moiré superlattices emerge with extended periodic interlayer modulation, inducing dramatic changes in electronic properties. Such engineered structures have unveiled a rich landscape of quantum phases, including unconventional superconductivity, correlated insulating states, ferroelectricity, ferromagnetism, and non-trivial topology, making moiré materials a highly tunable platform for exploring new physics and potential applications in quantum devices.

Computational Challenges of Modeling Moiré Superlattices

Computational Limits of Modeling Moiré Materials

While density functional theory (DFT) remains the cornerstone approach for electronic structure calculations, its computational cost limits direct simulations of large moiré-scale systems. Continuum models and tight-binding approaches that fit parameters to DFT data have been developed to capture the essential low-energy physics at significantly reduced cost.

However, their parametrisation and fixed functional form limit transferability. For example, for twisted bilayer transition metal dichalcogenides (TMDCs), continuum model parameters vary substantially across works targeting different twist angles and physical phenomena, restricting their predictive power for unexplored configurations or physics beyond the fitted data.

Addressing Transferability Gaps in Quantum Simulation

Addressing Transferability Gaps in Quantum Simulation

Tight-binding models also often require refitting when applied to different stacking configurations or materials. These limitations highlight the advantages that an ab initio-based efficient method for accurate large-scale moiré simulation could have. Machine learning (ML) techniques which predict a system’s electronic structure offer a promising path forward by learning directly from complete ab initio data and predicting unseen structures, avoiding the fixed analytical forms and the fitting limitations of traditional models, while maintaining access to the full set of ground state electronic properties.

Machine Learning for Advanced Electronic Structure Calculations

Machine Learning Approaches for Electronic Structure Calculations

For DFT-based Methods (ML-DFT), two main ML approaches have been explored: Hamiltonian-based Methods that directly predict real-space Hamiltonian matrix elements, and density-based Methods that predict the electron density distribution and derive electronic properties. State-of-the-art Hamiltonian-based Methods achieved impressive sub-meV accuracy for Hamiltonian matrix elements in training datasets of 2D materials and medium-sized twisted bilayers.

They exploit the nearsightedness principle of electronic matter to allow efficient sub-quadratic scaling with system sizes. This assumption can limit their applicability to systems with important long-range interactions, such as systems with weak electronic screening, significant Coulomb interactions, and charge rearrangements.

Direct prediction of the electron density offers an alternative ML approach, since the density uniquely determines all ground-state properties in the DFT formalism. In particular, using the density fitting (DF) technique to describe the electron density, as opposed to representing it on a grid, has been shown to yield a storage-efficient and transferable model with excellent accuracy in representing both core and bonding regions of real space.

Still, the challenge of the locality assumption for representing atomic environments persists, and the density-fitting approach can introduce noise which further complicates the intricate balance between density accuracy in training and the accuracy of downstream electronic properties. Their work addresses these challenges by extending the SALTED (Symmetry-Adapted Learning of Three-dimensional Electron Densities) model, which leverages a density fitting technique, an equivariant kernel method, and optimised Gaussian process regression with prediction cost linear in training set size.

They introduce long-range descriptors to capture non-local structural features, a numerical stabilisation technique to reduce DF noise, and explicit validation of downstream electronic properties, which together ensure reliable extrapolation of the electronic structure of 2D twisted-bilayer (TB) materials, even down to small twist angles. By predicting the electron densities of five TB-2D materials (graphene, hexagonal boron nitride (hBN), TiS2, ZrS2, MoS2), they demonstrate that local descriptors fail to capture the long-range physics of moiré systems, while long-range descriptors extending beyond the locality assumption enable accurate extrapolation of band-structures and electrostatic properties to large twisted superlattices.

They achieve robust predictions, with low-energy band error below 5 meV on twisted bilayer structures containing more than 1000 atoms. The resulting SALTED models successfully predict flat-band formation, spin-orbit coupling effects, and real-space observables, while achieving speedup between one and two orders of magnitude over fully converged DFT calculations.

This framework thus provides a generalisable approach for ML-accelerated electronic structure prediction in systems dominated by long-range interactions. Their approach is promising for accelerating the design of quantum materials and expanding the scope of first-principles simulations for complex quantum phenomena.

They first evaluated the ability of SALTED models to predict the electronic structure of 2D materials by applying them to validation sets of aligned (displaced) bilayer structures for each 2D bilayer material (graphene, hBN, TiS2, ZrS2, and MoS2). To enable controlled comparisons, they fixed all descriptor hyperparameters across experiments, varying only the GPR regularisation η.

The descriptors considered comprise the short-range SOAP, and the long-range LODE and LOVV descriptors. Table I summarises the accuracy of SALTED models for the electron density (direct output) and band structure (derived output). Definitions of the error metrics used can be found in Section IV D.

Consistent with previous work, they observed that the density error does not reliably correlate with the error in the prediction of derived observables, in this case the band structure. For graphene and ZrS2, the errors in both the density and band structure follow the more common relation: Descriptors that yield larger errors for density also result in larger errors on the band structure.

However, for TiS2 and MoS2, SOAP descriptors consistently achieve low density error, yet fail catastrophically in band structure predictions, while LOVV performs very well. For hBN and TiS2 a larger error in the density is also seen when using LODE, in comparison to SOAP, but smaller errors are observed in the band structure.

These results demonstrate that descriptor-dependent performance exists. Figure 1(A) illustrates the dependence of the accuracy of the extrapolated band structures on both the choice of descriptor and the system size. In Figure 1(A), they plot the low-energy moiré band error against the superlattice size for each material.

For graphene, all descriptors maintain reasonable accuracy ( 1300 atoms per superlattice. Only the graphene prediction is based on SOAP; the others are based on LOVV, and the predictions accurately match DFT band structures across all materials, demonstrating the framework’s capability to handle various 2D systems.

Validation of SALTED models for density and band structure prediction in 2D materials

Density prediction errors averaged 3.63% for graphene, 2.12% for hexagonal boron nitride, 2.12% for titanium disulphide, 0.45% for zirconium disulphide, and 0.04% for molybdenum disulphide, as determined by evaluating SALTED models on validation sets of aligned bilayer structures. Corresponding band structure errors, measured in meV, were 11, 4.5, 6.0, 10.1, and 15.9 for the same materials, respectively.

These initial validations demonstrate the potential of SALTED to accurately predict both electron densities and band structures in two-dimensional materials. The study revealed a nuanced relationship between density and band structure fidelity, where perfect density prediction does not guarantee accurate band structures, but error distribution significantly impacts electronic structure accuracy.

For graphene and zirconium disulphide, errors in both density and band structure aligned with expectations, with descriptors yielding larger density errors also resulting in larger band structure errors. However, titanium disulphide and molybdenum disulphide showed inconsistencies, with SOAP descriptors achieving low density error but failing catastrophically in band structure predictions, while LOVV performed well.

This behaviour highlights the importance of descriptor choice in accurately capturing the physics governing observable properties. Further assessment focused on extrapolating to small twist angles, requiring supercells ranging from 6 to 45 Å for graphene and hexagonal boron nitride, and 9 to 50 Å for transition metal dichalcogenides, containing up to 1324 atoms.

Low-energy moiré band errors were plotted against superlattice size, revealing that for graphene, all descriptors maintained reasonable accuracy below 5 meV even at large system sizes. However, SOAP and LODE exhibited degraded accuracy with increasing moiré length scales for other materials, with SOAP failing completely for titanium disulphide and molybdenum disulphide, while LOVV consistently maintained robust extrapolation capabilities. The framework accurately matched DFT band structures across all materials, demonstrating its capability to handle diverse 2D systems and paving the way for accelerating the design of quantum materials.

Extrapolating electronic structure predictions across large moiré superlattices using Gaussian process regression

Researchers have developed a transferable machine learning framework for predicting electronic structures in large moiré superlattices, overcoming limitations associated with conventional methods. This approach utilises Gaussian process regression, specifically a SALTED model, trained on electron densities of small bilayer structures and then extrapolated to larger supercells representing small twist angles.

The methodology achieves accurate predictions for complex phenomena, including flat band formation, bandwidth narrowing, and domain-wall electric fields, in materials like twisted bilayer graphene, hexagonal boron nitride, and transition metal dichalcogenides. The significance of this work lies in its ability to perform ab initio calculations on systems previously inaccessible due to computational demands.

By training the model on smaller, tractable structures, accurate extrapolations to much larger supercells, up to approximately 100 Angstroms, are possible, achieving speed-ups of 10 to 100times compared to direct density-functional theory calculations. The success of the method relies on three key innovations: the use of long-range descriptors encoding distant geometric information, a systematic resolution of numerical instabilities in density fitting, and an understanding that the spatial distribution of density errors significantly impacts the accuracy of downstream electronic property predictions.

While acknowledging numerical instabilities within the SALTED implementation, the authors suggest future work could explore active learning strategies to improve model accuracy or transition to a neural network architecture. This framework offers a general methodology for electronic structure prediction in large-scale systems, facilitating materials screening and the discovery of novel correlated quantum phases.