Quantum distance and anomalous Landau levels of flat bands

Semiclassical quantization of electronic states under a magnetic field describes not only the Landau level spectrum but also the geometric responses of metals under a magnetic field. 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 particles. 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. The Landau levels of such a flat band develop in the empty region in which no electronic states exist in the absence of a magnetic field, and exhibit an unusual dependence on the Landau level index n, which results in anomalous orbital magnetic susceptibility. The total energy spread of the Landau levels of flat bands is determined by the geometry of the relevant Bloch states, which is characterized by their Hilbert–Schmidt quantum distance. The results indicate that the anomalous Landau level spectrum of flat bands is promising for the direct measurement of the geometry of wavefunctions in condensed matter.