Quantum distance and anomalous Landau levels of flat bands.

Semiclassical quantization of electronic states under a magnetic field, as proposed by Onsager, describes not only the Landau level spectrum but also the geometric responses of metals under a magnetic field1-5. Even in graphene with relativistic energy dispersion, Onsager's rule correctly describes the π Berry phase, as well as the unusual Landau level spectrum of Dirac particles6,7. However, it is unclear whether this semiclassical idea is valid in dispersionless flat-band systems, in which an infinite number of degenerate semiclassical orbits are allowed. Here we show that the semiclassical quantization rule breaks down for a class of dispersionless flat bands called 'singular flat bands'8. The singular flat band has a band crossing with another dispersive band that is enforced by the band-flatness condition, and shows anomalous magnetic responses. The Landau levels of a singular flat band develop in the empty region in which no electronic states exist in the absence of a magnetic field, and exhibit an unusual 1/n dependence on the Landau level index n, which results in diverging orbital magnetic susceptibility. The total energy spread of the Landau levels of a singular flat band is determined by the quantum geometry of the relevant Bloch states, which is characterized by their Hilbert-Schmidt quantum distance. We show that there is a universal and simple relationship between the total Landau level spread of a flat band and the maximum Hilbert-Schmidt quantum distance, which can be verified in various candidate materials. The results indicate that the anomalous Landau level spectrum of flat bands is promising for the direct measurement of the quantum geometry of wavefunctions in condensed matter.