  We theoretically investigate the exchange interaction between magnetic
impurities in ultrathin Bi$_2$Se$_3$ topological insulator films by taking into
account the low-energy states produced by the impurities. We find that the
locally induced impurity resonances strongly influence the exchange interaction
between magnetic moments. In particular, we find a non-collinear alignment
being more favorable than the collinear ferromagnetic alignment preferred when
impurity states are ignored and only the pristine topological insulator band
structure is considered. Moreover, we show that by applying of an electric
field perpendicular to the ultrathin film, the exchange interaction can be
drastically enhanced. This opens for the possibility of highly tunable
magnetism by electric field.
