Variation of Landau level splitting in the Fermi level controlled Dirac metals (Eu,Gd)MnBi2

We have experimentally studied the Landau levels near the quantum limit in the magnetic Dirac material ($\mathrm{Eu},\mathrm{Gd}){\mathrm{MnBi}}_{2}$. In this series of materials, the Fermi level is systematically controlled by substituting ${\mathrm{Eu}}^{2+}$ with ${\mathrm{Gd}}^{3+}$ while keeping high mobility. We measured the Shubnikov--de Haas (SdH) oscillation for a single crystal with the lowest hole concentration at tilted magnetic fields up to 20--50 T and clarified the dependence of the splitting of the Landau levels $N\ensuremath{\ge}1$ on the ratio of Zeeman energy to cyclotron energy. In the low-field antiferromagnetic phase, the splitting is well explained by the (Zeeman) spin splitting, from which we have found that the effective $g$ factor of the Dirac fermion depends significantly on the Fermi energy. In the high-field antiferromagnetic phase, on the other hand, the SdH oscillation was found to change in a complex manner as a function of the tilt angle of the field, implying the lifting of the valley degeneracy as well as the spin degeneracy.