Time-reversal-breaking induced quantum spin Hall effect

We show that quantum spin Hall (QSH) effect does not occur in a square lattice model due to cancellation of the intrinsic spin-orbit coupling coming from different hopping paths. However, we show that QSH effect can be induced by the presence of staggered magnetic fluxes alternating directions square by square. When the resulting Peierls phase takes a special value gamma = pi/4, the system has a composite symmetry Theta P_pi/2 with Theta the time-reversal operator and P- pi/2 transforming the Peierls phase from gamma to gamma - pi/2, which protects the gapless edge states. Once the phase deviates from gamma = pi/4, the edge states open a gap, as the composite symmetry is broken. We further investigate the effect of a Zeeman field on the QSH state, and find that the edge states remain gapless for gamma = pi/4. This indicates that the QSH effect is immune to the magnetic perturbation.