Quantum distance and anomalous Landau levels of flat bands

Semiclassical quantization of electronic states under magnetic field, advocated by Onsager, can successfully describe not only the Landau level spectrum but also the geometric responses under magnetic field in metals. Even in graphene with relativistic energy dispersion, for instance, Onsager’s rule correctly described the π-Berry phase as well as the unusual Landau level spectrum of Dirac particles. Here we show that, however, the semiclassical quantization rule completely breaks down for a class of dispersionless flat bands, dubbed the singular flat bands. The singular flat band has a band crossing with another dispersive band enforced by band flatness. In contrast to the conventional isolated flat band which does not respond to external magnetic field, the singular flat band shows anomalous Landau level structures and magnetic responses. We discuss the novel quantum geometric interpretations of them regarding the quantum distance, and how to measure them experimentally.