  Using first principles methods, we investigate topological phase transitions
as a function of exchange field in a Bi(111) bilayer. Evaluation of the spin
Chern number for different magnitudes of the exchange field reveals that when
the time reversal symmetry is broken by a small exchange field, the system
enters the time-reversal broken topological insulator phase, introduced by Yang
{\it et al.} in Phys. Rev. Lett. 107, 066602 (2011). After a metallic phase in
the intermediate region, the quantum anomalous Hall phase with non-zero Chern
number emerges at a sufficiently large exchange field. We analyze the phase
diagram from the viewpoint of the evolution of the electronic structure, edge
states and transport properties, and demonstrate that different topological
phases can be distinguished by the spin-polarization of the edge states as well
as spin or charge transverse conductivity.
