We study the thermodynamic properties of the two-component $2+1$-dimensional massive Dirac fermions in an external magnetic field. The broken time-reversal symmetry results in the presence of a linear in the magnetic field part of the thermodynamic p
otential, while in the famous problem of Landau diamagnetism the leading field dependent term is quadratic in the field. Accordingly, the leading term of the explicitly calculated magnetization is anomalous, viz. it is independent of the strength of the magnetic field. The Stv{r}eda formula is employed to describe how the anomalous magnetization is related to the anomalous Hall effect.
A new family of the low-buckled Dirac materials which includes silicene, germanene, etc. is expected to possess a more complicated sequence of Landau levels than in pristine graphene. Their energies depend, among other factors, on the strength of the
intrinsic spin-orbit (SO) and Rashba SO couplings and can be tuned by an applied electric field $E_z$. We studied the influence of the intrinsic Rashba SO term on the energies of Landau levels using both analytical and numerical methods. The quantum magnetic oscillations of the density of states are also investigated. A specific feature of the oscillations is the presence of the beats with the frequency proportional to the field $E_z$. The frequency of the beats becomes also dependent on the carrier concentration when Rashba interaction is present allowing experimental determination of its strength.
