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. The semiclassical quantization rule breaks down for a class of dispersionless flat bands, and their anomalous Landau level spectrum is characterized by their Hilbert–Schmidt quantum distance.

中文翻译：

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平带的量子距离和反常朗道能级

正如 Onsager 所提出的，磁场下电子态的半经典量子化不仅描述了朗道能级谱，还描述了金属在磁场下的几何响应1-5。即使在具有相对论能量色散的石墨烯中，Onsager 的规则也正确地描述了 π Berry 相，以及 Dirac 粒子的不寻常的朗道能级谱 6,7。然而，尚不清楚这种半经典思想在无色散平带系统中是否有效，其中允许无限数量的简并半经典轨道。在这里，我们展示了半经典量化规则分解为一类称为“奇异平坦带”8 的无色散平坦带。奇异平坦带具有与另一个由带平坦度条件强制执行的色散带交叉的带，并显示异常的磁响应。奇异平带的朗道能级在没有磁场的情况下不存在电子态的空白区域发展，并表现出对朗道能级指数 n 的不寻常的 1/n 依赖性，这导致了发散的轨道磁化率。奇异平带的朗道能级的总能量扩展由相关布洛赫态的量子几何决定，其特征在于它们的希尔伯特-施密特量子距离。我们表明，平带的总朗道能级扩展与最大希尔伯特-施密特量子距离之间存在普遍而简单的关系，这可以在各种候选材料中得到验证。结果表明，平带的异常朗道能级谱有望用于直接测量凝聚态中波函数的量子几何。对于一类无色散平坦带，半经典量化规则被打破，它们的异常朗道能级谱以它们的希尔伯特-施密特量子距离为特征。