Topological phase transition in the antiferromagnetic topological insulator MnBi$$_2$$Te$$_4$$ from the point of view of axion-like state realization

This work aims to study the conditions of topological phase transition (TPT) between the topological and trivial states in the antiferromagnetic topological insulator (AFM TI) MnBi$$_2$$ 2 Te$$_4$$ 4 and propose some theory about the relationship of this TPT with the possibility of axion-like state realization in this material. Using the density functional approach we have analyzed the changes in the electronic and spin structure of topological surface states (TSSs) and the nearest conduction and valence bands (CB and VB) including the changes in the bulk band gap as well as the Dirac point (DP) gap in TSSs under variation of the spin-orbit coupling strength in the region of the TPT for infinite crystal and slab with a surface both. We have shown that in both cases the TPT occurs with inversion of the contributions of the Bi-$$p_z$$ p z and Te-$$p_z$$ p z states of different parity at the gap edges related to change in the gap sign. In the case of surface calculations, the Bi-$$p_z$$ p z and Te-$$p_z$$ p z states at the edges of the bulk band gap and their inversion at the TPT point are transformed into the TSSs with an energy gap at the DP. In this case the TPT takes place without closing the band gap, i.e. with a “jump” through zero and the formation of the nonzero gap during such a transition. Our calculations show that the TPT point is also characterized by an inversion of the out-of-plane spin polarization $$s_z$$ s z at the $$\Gamma$$ Γ point for lower and upper parts of the Dirac cone and a significant spatial redistribution of the TSSs between the surface and the bulk. We suppose that the nonzero Dirac gap can have some relationship with the formation of the axion-like state, which presumably couples nonmagnetic spin-orbit and magnetic contributions at the boundary between the topological and trivial phases for a system with parameters close to the TPT conditions. A practically realized system is proposed - the AFM TI with a stoichiometry close to that of MnBi$$_2$$ 2 Te$$_2$$ 2 Se$$_2$$ 2 with partial (about 50%) substitution of Te atoms for Se atoms in MnBi$$_2$$ 2 Te$$_4$$ 4 which can be an experimental platform for the implementation and experimental analysis of the TPT and the corresponding possibility of the axion-like state realization in Condensed Matter. Besides, such system could serve as a good platform for studying the dynamic axion effect, where the axion field fluctuations are maximised when a small external field is applied to the system which state is close to the TPT.