Quantum spin Hall effect induced by nonmagnetic and magnetic staggered potentials

We conducted a comparative study of the quantum spin Hall (QSH) effects induced by nonmagnetic and magnetic staggered potentials, respectively, and show that they have the same effect in driving the topological phase transition. The result implies that both time-reversal (T) preserving and breaking systems can host a QSH effect. We also investigate the stability of the resulting QSH effect for disorder and find that, for T invariant systems, the edge states are always robust while those of the T breaking system are also robust if there is additional symmetry in the system.