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

Quantum geometry of electronic state in momentum space, distinct from real-space structural geometry, has attracted increasing interest to shed light on understanding quantum phenomena. An interesting recent study [Nature 584, 59-63 (2020)] has numerically solved a 2-band effective Hamiltonian to show the anomalous Landau level (ALL) spreading $\mathit{\Delta}$ of a singular flat band (SFB), such as hosted in a kagome lattice, in relation to the maximal quantum distance $d$ of the SFB, $\mathit{\Delta}(d)$, which enables a direct measure of quantum geometry. Here, we investigate the ALLs of SFB by studying both the 2-band Hamiltonian and a diatomic kagome lattice hosting two SFBs mirrored by particle-hole symmetry. We derive an exact analytical solution of the 2-band Hamiltonian to show there are two branches of $\mathit{\Delta}(d)$. Strikingly, for the diatomic kagome lattice, $\mathit{\Delta}$ depends on not only $d$ but also $r$, the real-space diatomic distance. As $r$ increases, $\mathit{\Delta}$ shrinks toward zero while $d$ remains intact, which can be intuitively understood from the magnetic-field-induced disruption of destructive interference of the SFB compact localized states. Based on semiclassical theory, we derive rigorously the dependence of $\mathit{\Delta}$ on $r$ that originates from the tuning of the non-Abelian orbital moment of the two SFBs by real-space geometry.