Quantum geometry, flat Chern bands, and Wannier orbital quantization

Dispersionless (flat) quantum states is a paradigm shift in condensed matter physics, serving as a natural platform for emergent electronic phases. In this work, we propose the foundation for the new framework addressing the issue of band flatness from quantum-geometrical perspective involving Wannier functions probes in the real space. The perfectly flat bands serve as ideal objects for classification, ranging from intrinsically distinct examples such as flat electronic bands in argon ice, artificial lattices (Kagome, Lieb), Landau levels, and magic-angle twisted bilayer graphene and its descendants. Through using the quantum geometry concepts and Wannier function analysis we reveal a hidden connection between the flatness of topological bands, their Chern numbers, and quantum-geometrical limit for flat-band orbital cross section, reminiscent of Lifshitz-Onsager quantization.