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.
Onsager, L. Interpretation of the de Haas–van Alphen effect. Philos. Mag. 43, 1006–1008 (1952).
Roth, L. M. Semiclassical theory of magnetic energy levels and magnetic susceptibility of Bloch electrons. Phys. Rev. 145, 434–448 (1966).
Mikitik, G. P. et al. Manifestation of Berry’s phase in metal physics. Phys. Rev. Lett. 82, 2147–2150 (1999).
Gao, Y. & Niu, Q. Zero-field magnetic response functions in Landau levels. Proc. Natl Acad. Sci. USA 114, 7295–7300 (2017).
Fuchs, J.-N. et al. Landau levels, response functions and magnetic oscillations from a generalized onsager relation. SciPost Phys. 4, 024 (2018).
Zhang, Y. et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nat. Phys. 2, 177–180 (2006).
Rhim, J.-W. & Yang, B.-J. Classification of flat bands according to the band-crossing singularities of Bloch wave functions. Phys. Rev. B 99, 045107 (2019).
Bužek, V. & Hillery, M. Quantum copying: beyond the no-cloning theorem. Phys. Rev. A 54, 1844–1852 (1996).
Dodonov, V. V. et al. Hilbert–Schmidt distance and non-classicality of states in quantum optics. J. Mod. Opt. 47, 633–654 (2000).
Berry, M. V. in Geometric Phases in Physics (eds Shapere, A. & Wilczek, F.) 7–28 (World Scientific, 1989).
Haldane, F. D. M. Dirac-point models: Hilbert space geometry and topology http://wwwphy.princeton.edu/~haldane/talks/nobel_jpeg.pdf (2010).
Neupert, T. et al. Measuring the quantum geometry of Bloch bands with current noise. Phys. Rev. B 87, 245103 (2013).
Peotta, S. et al. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).
Piéchon, F. et al. Geometric orbital susceptibility: quantum metric without Berry curvature. Phys. Rev. B 94, 134423 (2016).
Gianfrate, A. et al. Measurement of the quantum geometric tensor and of the anomalous Hall drift. Nature 578, 381–385 (2020).
Ozawa T. & Goldman N. Extracting the quantum metric tensor through periodic driving. Phys. Rev. B 97, 201117 (2018).
Park, S. & Yang, B.-J. Classification of accidental band crossings and emergent semimetals in two dimensional noncentrosymmetric systems. Phys. Rev. B 96, 125127 (2017).
Yang, B.-J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2014).
Xiao, Y. et al. Landau levels in the case of two degenerate coupled bands: kagome lattice tight-binding spectrum. Phys. Rev. B 67, 104505 (2003).
Yamada M. G. et al. First-principles design of a half-filled flat band of the kagome lattice in two-dimensional metal–organic frameworks. Phys. Rev. B 94, 081102 (2016).
Chen, Y. et al. Ferromagnetism and Wigner crystallization in kagome graphene and related structures. Phys. Rev. B 98, 035135 (2018).
You, J.-Y. et al. Flat band and hole-induced ferromagnetism in a novel carbon monolayer. Sci. Rep. 9, 20116 (2019).
Lee, J. M. et al. Stable flatbands, topology, and superconductivity of magic honeycomb networks. Phys. Rev. Lett. 124, 137002 (2020).
Ye, L. et al. Massive Dirac fermions in a ferromagnetic kagome metal. Nature 555, 638–642 (2018).
Lin, Z. et al. Flatbands and emergent ferromagnetic ordering in Fe3Sn2 kagome lattices. Phys. Rev. Lett. 121, 096401 (2018).
Kang, M. et al. Dirac fermions and flat bands in the ideal kagome metal FeSn. Nat. Mater. 19, 163–169 (2020).
Kang, M. et al. Topological flat bands in frustrated kagome lattice CoSn. Preprint at https://arxiv.org/abs/2002.01452 (2020).
Li, Z. et al. Realization of flat band with possible nontrivial topology in electronic kagome lattice. Sci. Adv. 4, eaau4511 (2018).
Yin, J.-X. et al. Negative flat band magnetism in a spin–orbit-coupled correlated kagome magnet. Nat. Phys. 15, 443–448 (2019).
Min, H. et al. Intrinsic and Rashba spin–orbit interactions in graphene sheets. Phys. Rev. B 74, 165310 (2006).
Ramachandran A. et al. Chiral flat bands: existence, engineering, and stability. Phys. Rev. B 96, 161104 (2017).
Ihn, T. Semiconductor Nanostructures: Quantum States and Electronic Transport (Oxford Univ. Press, 2010).
Terashima, T. T. et al. Magnetization process of the Kondo insulator YbB12 in ultrahigh magnetic fields. J. Phys. Soc. Jpn. 86, 054710 (2017).
Mayorov, A. S. et al. Micrometer-scale ballistic transport in encapsulated graphene at room temperature. Nano Lett. 11, 2396–2399 (2011).
Stoner, E. Atomic moments in ferromagnetic metals and alloys with nonferromagnetic elements. Phil. Mag. 15, 1018–1034 (1933).
Kopnin N. P. et al. High-temperature surface superconductivity in topological flat-band systems. Phys. Rev. B 83, 220503 (2011).
Hanaguri T. et al. Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3. Phys. Rev. B 82, 081305 (2010).
Sadowski, M. L. et al. Landau level spectroscopy of ultrathin graphite layers. Phys. Rev. Lett. 97, 266405 (2006).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Perdew, J. P. et al. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Dudarev, S. L. et al. Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys. Rev. B 57, 1505–1509 (1998).
Po, H. C. et al. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).
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