Abstract
When topological insulators meet broken time-reversal symmetry, they bring forth many novel phenomena, such as topological magnetoelectric, half-quantum Hall, and quantum anomalous Hall effects. From the well-known quantum spin Hall state in Bi${}_{2}$Se${}_{3}$ thin films, we predict various topological and magnetic phases when the time-reversal symmetry is broken by magnetic ion doping. As the magnetic ion density increases, the system undergoes successive topological or magnetic phase transitions due to variation of the exchange field and the spin-orbit coupling. In order to identify the topological phases, we vary the spin-orbit coupling strength from zero to the original value of the system and count the number of band crossings between the conduction and valence bands, which directly indicates the change of the topological phase. This method provides a physically intuitive and abstract view to figure out the topological character of each phase and the phase transitions between them.
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