标题： 奇异平带中异常朗道能级的量子与实空间几何相互作用 显示英文标题

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

摘要： 动量空间中电子态的量子几何，不同于实空间的结构几何，正引起越来越多的关注，以帮助理解量子现象。 一项有趣的研究 [Nature 584, 59-63 (2020)] 数值求解了一个两带有效哈密顿量，以展示奇异平带（SFB）的异常朗道能级（ALL）扩展$\mathit{\Delta}$，例如在凯格姆晶格中所具有的，这与SFB的最大量子距离$d$相关，$\mathit{\Delta}(d)$，从而可以直接测量量子几何。 在这里，我们通过研究两带哈密顿量和一个包含两个由粒子-空穴对称性镜像的奇异平带的二原子凯格姆晶格来研究SFB的ALL。 我们推导出两带哈密顿量的精确解析解，以显示存在两个分支的$\mathit{\Delta}(d)$。 引人注目的是，对于二原子凯格姆晶格，$\mathit{\Delta}$不仅取决于$d$，还取决于$r$，即实空间的二原子距离。 随着$r$增加，$\mathit{\Delta}$趋近于零，而$d$保持不变，这可以从磁场所诱导的 SFB 紧致局域态的破坏性干涉的破坏中直观理解。基于半经典理论，我们严格推导了$\mathit{\Delta}$对$r$的依赖关系，该依赖关系来源于实空间几何对两个 SFB 的非阿贝尔轨道矩的调节。 显示英文摘要

摘要： 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.