Cyclic structure of Landau levels in transition metal dichalcogenide semiconductors

Abstract: Transition metal dichalcogenides (TMDs) exhibit unconventional Landau level (LL) spectra that cannot be fully captured by an effective mass approximation or a minimal two-band Dirac model. Namely, TMDs show an anomalous, upward-sloping zeroth LL in the valence band and an asymmetric orbital magnetization between electron and hole bands. In this paper, we employ a continuum three-band model to derive analytic constraints on the LL spectrum of the $K$ and $K'$ valleys at weak magnetic fields. This model highlights the cyclic structure of the LL spectrum inherited from $C_3$ symmetry, providing both analytical tractability and an accurate description of the band geometry in the low energy approximation of the valleys. We compare our results against numerical calculations using the three-band tight-binding model of Ref.[1] and a distorted kagome lattice model. We find that the Landau levels of the $K$ and $K'$ valleys show a cyclic structure which explains their anomalous slope and magnetization asymmetry. This asymmetry can be traced to the topological obstruction of TMD semiconductors. We further analyze the impact of disorder, finding that the zeroth LL exhibits partial robustness against certain off-diagonal perturbations, in contrast to the exact index-theorem protection of massive Dirac particles. Our results establish a direct link between orbital structure, band topology, and magnetic response in TMDs.