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

TL;DR

This paper explores how quantum and real-space geometries influence anomalous Landau levels in singular flat bands, revealing new dependencies and analytical solutions that deepen understanding of quantum phenomena in lattice systems.

Contribution

It provides an exact analytical solution for the Landau level spreading in singular flat bands and uncovers the influence of real-space geometry on quantum states.

Findings

Landau level spreading depends on quantum distance and real-space diatomic distance.

Increasing diatomic distance reduces Landau level spreading, independent of quantum distance.

Real-space geometry tunes the non-Abelian orbital moment of flat band states.

Abstract

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.