6 October 2020 Tue 4 pm

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Since the recent discovery of the superconductivity in the twisted bilayer graphene, flat band systems have been studied intensively. We uncover the novel roles of the band-crossing singularities in flat bands. While the singularity of the flat band is topologically trivial, it can be characterized by a pseudo-spin canting around the touching point. We propose that the strength of the singularity is defined as the maximum canting angle between two pseudo-spins, or equivalently, the maximum quantum distance between two Bloch wave functions. This bulk’s singularity guarantees the existence of two topological objects in real space, one is the non-contractible loop state under the periodic boundary condition, and the other is the robust boundary modes under the open boundary condition. Then, we show that when the flat band has a singularity, its Landau levels develop in the energy gap, and their spreading into the gap is determined by the maximum quantum distance around the band crossing point. This is a new rule for the Landau level quantization, and completely beyond the conventional semiclassical paradigm. Moreover, the Landau levels corresponding to the flat band exhibit 1/n dependence, where n is the Landau level index. This behavior leads to the divergence of the orbital magnetic susceptibilities.