Topological metallic phases in spin–orbit coupled bilayer systems

We investigate the influence of different spin–orbit couplings on topological phase transitions in the bilayer Kane–Mele model. We find that the competition between intrinsic spin–orbit coupling and Rashba spin–orbit coupling can lead to two dimensional topological metallic states with nontrivial topology. Such phases, although having a metallic bulk, still possess edge states with well defined topological invariants. Specifically, we show that with preserved time reversal symmetry the system can exhibit a {Z₂} -metallic phase with spin helical edge states and a nontrivial {Z₂} invariant. When time reversal symmetry is broken, a Chern metallic phase could appear with chiral edge states and a nontrivial Chern invariant.