Model study and numerical calculation

This paper introduces representative theoretical models and calculation methods that are central to the study of topological phases and quantum geometric effects. Specifically, the Hofstadter butterfly model unravels the fractal electronic spectrum and its intrinsic connection with geometric phases under strong magnetic fields. Honeycomb lattice-based models, including graphene, Haldane, and Kane-Mele models, have demonstrated the existence of quantum anomalous Hall effects and symmetry-protected topological phases. Low-energy continuum models provide a valuable analytical understanding of topological phases, phase transitions, and edge-mode formation. First-principles calculations combined with Wannier function interpolation enable quantitative calculation of Berry curvatures and anomalous transport in real materials. Disorder models elucidate the robustness of topological phases, disorder-induced topological phase transitions, and Chern number annihilation mechanisms. We emphasize the fundamental importance of theoretical models and computational methods in developing the theory of quantum geometry and topological materials.