Data-driven approach and analysis for electronic structure in moiré materials.

Abstract

Quantum materials are characterized by complex behaviors that cannot be fully explained by traditional theories, requiring new approaches to understand their electronic properties. Among these, moiré materials—--constructed by stacking monolayer periodic sheets—--have demonstrated the potential for groundbreaking discoveries, particularly in the context of superconductivity at low temperatures. The case of twisted bilayer graphene (TBG) has become a cornerstone example, driving intense theoretical and experimental investigations in moiré materials including transition-metal dichalcogenides (TMD). Part I--Continuum Model for TMD The fundamental approach to understanding the electronic properties of materials involves time-dependent many-body Schrödinger equations describing the quantum dynamics of electrons and ions. However, the computational cost of simulating the many-body model exceeds the capabilities of current computing resources. One way to address this computational challenge is by using reduced-order models, including the tight-binding model, continuum models, and mean-field models (such as Hartree-Fock and Kohn-Sham DFT). In Part I of the thesis, we develop a continuum model for transition metal dichalcogenide (TMD) bilayers and investigate its limitations compared to the tight-binding model through rigorous numerical analysis. This continuum model can be used to compute various physical properties of interest, including the density of states, local density of states, and electrical conductivity. Moreover, it contributes to the development of quantum many-body models for flat bands. Unlike twisted bilayer graphene, the most intriguing electronic properties of bilayer TMD materials are primarily associated with a specific layer within the bilayer system. A useful perspective for understanding the electronic structure of TMDs is to draw an analogy with open quantum systems: one layer acts as the primary system, while the other serves as an environment influencing its electronic properties. We propose one approach to construct the continuum model for the layer exhibiting the most interesting electronic behavior. To illustrate our idea, we consider a coupled chain model where the top layer and bottom layer have different lattices and electronic properties is mainly related to one specific layer. The model serves as a 1-D counterpart for twisted bilayer TMD materials. We start with a tight-binding model in which the coupling between two orbitals depends not only on their spatial separation but also on the local environment of each orbital pair. Part II--Operator Learning for Moiré Materials In Part II of the thesis, to identify new moiré materials that may exhibit superconductivity, we have developed a data-driven computational framework that utilizes AI techniques to predict the local density of states (LDOS) of these materials. The LDOS provides critical insight into the electronic properties of solids and can be experimentally probed. Our approach leverages an operator learning-based method to identify singularities in the LDOS, which signal potential superconducting behavior. This method bypasses the need for resource-intensive aperiodic system simulations. We have formulated an inverse problem theory that underpins the effectiveness of this data-driven approach, proving that with high-quality input data and sufficient neural network training, we can achieve accurate predictions with rigorous numerical analysis. This research introduces a new paradigm for using machine learning in scientific computing, offering a powerful tool for the preliminary screening of Moire materials with superconducting potential. Part III--Quantum Dynamics for Incommensurate System In Part III of the thesis, We study the quantum dynamics of electrons in twisted bilayer graphene and general aperiodic system, modeled at the atomic scale using a tight-binding framework. For twisted bilayer graphene, a key challenge in such systems is the lack of periodicity for generic twist angles, which makes conventional computational methods inapplicable. In this work, we establish that the time evolution of the tight-binding model can be accurately approximated by computations on finite domains. The core of our analysis is a propagation speed bound derived via a Combes-Thomas estimate, which applies broadly to general aperiodic tight-binding systems. This result provides a rigorous foundation for simulating quantum dynamics in twisted bilayer graphene and offers new tools for analyzing transport and localization phenomena beyond periodic and quasicrystalline settings. Furthermore, we establish two Combes-Thomas estimate for general aperiodic system and general aperiodic system with low dimensional structure. In addition, we discussed the computational complexity for computing the quantum dynamics for aperiodic system. We complement our theoretical findings with numerical simulations that further illustrate the dynamical behavior of electrons in twisted bilayer graphene.