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Topological phases of Bi(111) bilayer in an external exchange field

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.