Moiré materials have raised significant interest in recent years due to the observa- tion of various intriguing phenomena, including novel superconductivity, the quantum anomalous Hall effect, and its fractional version. However, there are two significant challenges in the study of moiré materials. The first challenge is to accurately and efficiently present the low-energy band structure in such large systems. The second challenge is to find new materials where inhomogeneity caused by strain and twist has minimal impact on experiments.
In the first part of this dissertation, I discuss a newly developed numerical method called the Truncated Atomic Plane Wave (TAPW) method, which addresses the first challenge. For twisted bilayer graphene, this method serves as a natural extension of the original Bistritzer and MacDonald model, incorporating corrections for higher- order Fourier components and non-local tunneling, by projecting the full tight binding Hamiltonian on a series of plane waves. Additionally, we combine this method with numerical atomic orbital-based density functional theory to present the valley-resolved band structure of moiré transition metal dichalcogenide (TMD) materials. By drawing an analogy between the tight-binding formalism of electrons and phonons, we solve the problem of the low-frequency moiré phonons and systematically study the effective coupling between flat band electrons and transverse optical phonons in the magic-angletwisted bilayer graphene (MATBG) system.
The second part of this dissertation focuses on the second challenge, primarily proposing a new platform to engineer moiré physics. Here, I focus on another repre- sentation of low-dimensional materials: topological thin films, with a special emphasis on high quality Dirac semimetal Cd3As2. The magneto transport properties of Cd3As2 thin films were meticulously studied, revealing a possible Weyl semimetal phase and a quantum anomalous Hall phase both induced by in-plane Zeeman coupling. The (001) grown Cd3As2 thin film at critical thickness serves as an experimental realization of the Bernevig-Hughes-Zhang (BHZ) model. In light of recent experimental progress in superlattice and gate engineering, the emergent band topology in the BHZ model with superlattice potential is discussed, and an interaction driven Chern insulator phase can be available by spontaneously breaking time reversal symmetry.
