Quantum spin Hall effect induced by non-magnetic and magnetic staggered potentials

We have a comparative study of the quantum spin Hall (QSH) effects induced by non-magnetic 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 ($\mathcal{T}$) preserving and breaking systems can host QSH effect. We also investigate the stability of the resulting QSH effect to disorder and find that for $\mathcal{T}$ invariant system the edge states are always robust while those of $\mathcal{T}$ breaking system are also robust if there is additional symmetry in the system.