Topological Effects in Carbon-based Two-dimensional Nanostructures

Recent advances on low-dimensional and topological materials has greatly inspired the research in condensed matter physics. This thesis is devoted to the computational and theoretical study of topological effects in two-dimensional materials, especially nanostructures based on graphene. The theoretical research contained in the thesis is in different levels: minimal models and tight-binding modeling of materials. Electronic interactions are discussed effectively as well.In the first part of the thesis, I provide the theoretical study on a specific family of topological insulators: the Euler insulators, which are characterized by the Euler class. Noticing the relation between the mathematical expression of the Chern class and the Euler class, there rises the question about the edge states and magnetotransport features of Euler insulators. I show that the Euler insulators carry a series of signatures in their Landau levels. In contrast to trivial bands, the topological Euler bands exhibit a Landau level broadening under magnetic fields. Moreover, I found that the broadening becomes more significant in the case of larger Euler numbers. With the flat and degenerated bands serving as the simplest limit for investigating Euler insulators, I further unveil the edge signatures of Euler insulators. In the case of flat bands, the edge states of Euler insulators exhibit a series of crossings. The order of the crossings gives the largest Euler number, which can be seen from interpolating edge modes by a set of polynomials.The second part of the thesis presents a collection of research on topological effects in twisted multilayer graphene. The computational studies are performed using atomistic tight-binding (TB) models, providing the topological phase in twisted multilayer systems. I explore the topological phase and quantum geometric tensors of twisted bilayer graphene, and the interplay between band topology and spin textures. To compare with the transport measurements, I also provide the results of Hofstadter butterfly spectra for different topological phases. The idea of probing the flat-band topology with quantum Hall response is then verified in an example material: twisted double bilayer graphene. In addition to the Hofstadter spectra, I also discuss the topological effects on electronic interactions in a qualitative manner.Finally, I present the computational study of electronic transmission in wrinkled graphene sheets. By comparing the transmission in commensurate and incommensurate scenarios, I found a suppressed back-scattering in the incommensurate wrinkles. Therefore, I conclude that layer commensuration plays an important role in the transport of wrinkled 2D materials. The results provide guidelines to controlling the transport properties of graphene in presence of out-of-plane disorder.