Topological nature of the proper spin current and the spin-Hall torque

Spin currents are key to spin torque devices, but determining the proper spin current is nontrivial. Here we derive a general quantum-mechanical formula for the intrinsic proper spin current showing that it is topological and can be finite in the gap. For topological insulators with an out of plane magnetization and the chemical potential in the surface state gap, the net spin torque is determined by the competition between the topological spin-Hall torque due to the bulk and the topological Edelstein effect due to the surface states. We also discuss spin-3/2 hole quantum wells.