Hofstadter spectra for d-orbital electrons: A case study on MoS₂

To study the Hofstadter spectrum of monolayer molybdenum disulfide (MoS₂), we systematically examine the magnetic energy spectra due to various hoppings between d-orbital electrons to illustrate the link between the Bloch bands and the corresponding magnetoelectronic spectra. The magnetoelectronic spectrum shows a mirror (inversion) symmetry as a result of the particle-hole symmetry (asymmetry) in the Bloch bands. At small field, specific Landau fan diagrams can be ascribed to certain Bloch-band singularities. In the spectrum of real MoS₂, we further find a breaking of spectral symmetry, the spin and valley polarization, and a flux-dependent energy gap. Our numerical results can facilitate a qualitative understanding of the topological nature of d-bands and provide a basis for exploring the Landau levels in transition-metal dichalcogenides.