Variety of scenarios for magnetic exchange response in topological insulators

We present an ab initio relativistic k.p theory of the effect magnetic exchange field on band structure in gap region bulk crystals and thin films three-dimensional layered topological insulators. For perpendicular to layers (along $z$), we reveal novel unconventional scenarios response band-gap edges magnetization. The modification valence conduction states is considered terms their $\Gamma$-point spin $s^z$ total angular momentum $J^z$ atomic sites where are localized. actual scenario depends whether have same or opposite sign. In particular, sign for state give rise Bi$_2$Te$_3$ -- both crystal ultra-thin films, which fundamentally distinguishes this insulator from Bi$_2$Se$_3$, To gain a deeper insight into different insulators with inverted non-inverted zero-field structure, minimal four-band third-order model constructed first principles. Within model, analyze field-induced identify Weyl nodes that appear phase behave differently depending scenario. characterize topology modified by Chern number $\mathcal{C}$ find be accompanied large $\mathcal{C}=\pm3$.