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 field1,2,3,4,5. 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.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
The codes used to generate the data of the current study are available from the corresponding author on reasonable request.
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J.-W.R. was supported by the Institute for Basic Science (IBS-R009-D1). K.K. was supported by a National Research Foundation of Korea (NRF) grant (contract 2016R1D1A1B02008461) and the Internal R&D programme at KAERI (grant number 524210-20). B.-J.Y. was supported by the Institute for Basic Science (IBS-R009-D1), the Basic Science Research Program through the NRF (grant number 0426-20200003) and the US Army Research Office under grant number W911NF-18-1-0137.
The authors declare no competing interests.
Peer review information Nature thanks Justin C. W. Song and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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The file consists of 18 Supplementary Sections for details on solving Landau level problems, analytic proofs for the main results, various flat-band models, disorder problems, Landau level properties of nearly flat bands, Chern number calculations, and so on. 16 Supplementary Figures and one Table are included.
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Rhim, JW., Kim, K. & Yang, BJ. Quantum distance and anomalous Landau levels of flat bands. Nature 584, 59–63 (2020). https://doi.org/10.1038/s41586-020-2540-1
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