Temperature dependence of the spin-flop transition in a topological antiferromagnet MnBi$$_2$$Te$$_4$$

Experimental studies of magnetic properties of a topological van der Waals antiferromagnet insulator MnBi $$_2$$ Te $$_4$$ demonstrated not only an anomalous behavior of magnetization before and after the spin-flop transition but also its strong temperature dependence. To interpret these effects, we present a quantum theory of layered antiferromagnet with a trigonal symmetry of the triangular lattice. Using atomic representations for spin operators and the diagram technique for Hubbard operators, we obtain a dispersion equation describing the temperature dependence of the excitation spectrum. In the anisotropic self-consistent field approximation, we derive a transcendent equation establishing an interrelation between the Néel temperature and model parameters. For the weak-anisotropy case, we obtain its analytical solution. We describe the temperature evolution of magnetization as a function of magnetic field and construct a phase diagram showing regions where different configurations of MnBi $$_2$$ Te $$_4$$ magnetic sublattices are realized. We observe that quantum effects induced by the trigonal component of the single-ion anisotropy essentially affect the thermodynamic properties of antiferromagnets.