Interplay of quantum geometry and real-space geometry in the anomalous Landau levels of singular flat bands

The anomalous Landau levels (LLs) of flat bands (FBs) beyond the Onsager's semiclassical quantization rule have drawn significant attention recently. One most striking finding is that the energy spread of the LLs of a singular FB is determined by the quantum geometry of the FB wavefunction [Nature 584, 59-63 (2020)]. Here, we revisit this problem by studying the anomalous LLs of a diatomic kagome lattice that hosts two singular FBs. Surprisingly, we find that the anomalous LL spreading depends not only on quantum distance (d_Q) between FB states but also on real-space distance (d_R), the diatomic distance. Most interestingly, as d_R increases, the LL spread of a singular FB evolves towards zero even though the maximal d_Q remains unchanged. This evolution can be intuitively understood from the magnetic-field-induced disruption of destructive interference of the localized FB wavefunction, which is strongest at the smallest d_R, the limit of a kagome lattice. Furthermore, we derive an effective two-band continuum Hamiltonian to show that the dependence of anomalous LL spectrum on the real-space geometry is generally encoded in the coefficients of the commutator of lattice momenta k_x and k_y upon Peierls substitution. Our study highlights the physical manifestation of destructive interference and the non-commutative nature of k_x and k_y under magnetic field.