Interplay between magnetism and topology in MnBi$_2$Te$_4$

The dependence of the topology of the fermion excitation spectrum on the magnetic state of the system is analyzed taking into account the structure of the Te–Mn–Te trilayer in the Te–Bi–Te–Mn–Te–Bi–Te layer sequence of the MnBi$_2$Te$_4$ van der Waals single crystal, crystal field effects, spin–orbit coupling, and the covalent mixing of electronic states of Mn$^{2+}$ ions with electronic states of Te$^{2-}$ ions in the strong electron correlation regime. The Chern number in the ferromagnetic phase, which is due to the kinematic interaction between Hubbard fermions, is equal to 1; i.e., the topology of the band structure of the Te–Mn–Te trilayer is nontrivial. The Chern number in the paramagnetic phase is zero; i.e., the topology is trivial. The magnetic moments of Mn$^{2+}$ ions for the constructed spin orbitals are perpendicular to the layers. The magnetic moments of Mn$^{2+}$ ions in the nearest layers are antiferromagnetically ordered via the Anderson mechanism.