  A topological insulator doped with random magnetic impurities is studied. The
system is modelled by the Kane-Mele model with a random spin exchange between
conduction electrons and magnetic dopants. The dynamical mean field theory for
disordered systems is used to investigate the electron dynamics. The magnetic
long-range order and the topological invariant are calculated within the mean
field theory. They reveal a rich phase diagram, where different magnetic
long-range orders such as antiferromagnetic or ferromagnetic one can exist in
the metallic or insulating phases, depending on electron and magnetic impurity
fillings. It is found that insulator only occurs at electron half filling,
quarter filling and when electron filling is equal to magnetic impurity
filling. However, non-trivial topology is observed only in half-filling
antiferromagnetic insulator and quarter-filling ferromagnetic insulator. At
electron half filling, the spin Hall conductance is quantized and it is robust
against magnetic doping, while at electron quarter filling, magnetic dopants
drive the ferromagnetic topological insulator to ferromagnetic metal. The
quantum anomalous Hall effect is observed only at electron quarter filling and
dense magnetic doping.
