Hofstadter spectrum in MoS$_2$

In studying the Hofstadter problem of monolayer molybdenum disulfide (MoS$_2$), we systematically demonstrate the magnetic energy spectra due to various hoppings between $d$-orbital electrons. The magnetoelectronic spectrum shows a mirror symmetry as a result of the particle-hole symmetry in the Bloch bands. At small field, specific Landau fan diagrams can be ascribed to certain Bloch-band singularities. In the spectrum of real MoS$_2$, we further illustrate a breaking of spectral symmetry, the spin and valley polarization, and a flux-dependent energy gap. Our numerical results can facilitate the qualitative understanding of topological nature of $d$-bands and provide a basis for exploring the Landau levels in transition-metal dichalcogenides.