  Since the discovery of the quantum anomalous Hall effect in the magnetically
doped topological insulators (MTI) Cr:(Bi,Sb)$_2$Te$_3$ and
V:(Bi,Sb)$_2$Te$_3$, the search for the exchange coupling mechanisms underlying
the onset of ferromagnetism has been a central issue, and a variety of
different scenarios have been put forward. By combining resonant photoemission,
X-ray magnetic dichroism and multiplet ligand field theory, we determine the
local electronic and magnetic configurations of V and Cr impurities in
(Bi,Sb)$_2$Te$_3$. While strong pd hybridisation is found for both dopant
types, their 3d densities of states show pronounced differences.
State-of-the-art first-principles calculations show how these impurity states
mediate characteristic short-range pd exchange interactions, whose strength
sensitively varies with the position of the 3d states relative to the Fermi
level. Measurements on films with varying host stoichiometry support this
trend. Our results establish the essential role of impurity-state mediated
exchange interactions in the magnetism of MTI.
