Fluctuation Mechanism of Single-Ion Anisotropy of Topological Insulator MnBi$_2$Te$_4$

Abstract

We demonstrate that charge fluctuations induced by electron hopping, combined with spin-orbit coupling, lift the sixfold degeneracy of the orbital singlet $^{6}S$ of Mn ions in the topological insulator MnBi$_2$Te$_4$, resulting in single-ion anisotropy. To solve the problem, a multiplet representation is introduced for the creation operators of atomic-state fermions in terms of the operators describing transitions between many-body wavefunctions. Using the operator form of perturbation theory up to the second order, we derive expressions for the populations $n_M$ of Mn ion states with spin projections $M$ of the $^{6}S$ term and determine the single ion anisotropy constants. The calculations reveal that the fluctuation mechanism ensures the possibility of implementing the easy-axis anisotropy observed in MnBi$_2$Te$_4$. Notably, the range of anisotropy constants $D_2$ obtained by varying the model parameters includes the value $D_2 = -0.0095$ meV, required to reproduce the critical field of the spin-flop transition $H_{\text{sf}}$, known from the experiment. The proposed mechanism has a wide range of applicability for describing the anisotropy in compounds where the ground state of a magnetic ion in a weak crystal field is described by an orbital singlet.