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

Variety of scenarios of the magnetic exchange response in topological insulators