Anomalous Landau levels and quantum oscillation in rotation-invariant insulators

Landau levels in certain models are known to protrude into the zero-field energy gap. These are known as anomalous Landau levels (ALLs). We study whether ALLs can lead to Fermi-surface like quantum oscillation in the absence of a zero-field Fermi surface. Focusing on two-dimensional multi-band low-energy models of electrons with continuous rotation symmetry, we show that an effective-band description, akin to the semiclassical treatment of Landau level problems in metals, can be used to predict the Landau level spectrum, including possible ALLs. This description describes ALL induced quantum oscillation for certain insulating models, which we demonstrate through numerical calculations. We demonstrate that these results could guide the search for ALL in 2D topological materials with effective Hamiltonian similar to that of magnetically doped (Bi, Sb)2Te3 ultrathin films.