Topologically protected extended states in disordered quantum spin-Hall systems without time-reversal symmetry

Abstract

We demonstrate the existence of robust bulk extended states in the disordered Kane-Mele model with vertical and horizontal Zeeman fields in the presence of a large Rashba coupling. The phase diagrams are mapped out by using level statistics analysis and computations of the localization length and spin-Chern numbers ${C}_{\ifmmode\pm\else\textpm\fi{}}$. ${C}_{\ifmmode\pm\else\textpm\fi{}}$ are protected by the finite energy and spin mobility gaps. The latter stay open for arbitrarily large vertical Zeeman fields, or for horizontal Zeeman fields below a critical strength or at moderate disorder. In such cases, a change of ${C}_{\ifmmode\pm\else\textpm\fi{}}$ is necessarily accompanied by the closing of the mobility gap at the Fermi level. The numerical simulations reveal sharp changes in the quantized values of ${C}_{\ifmmode\pm\else\textpm\fi{}}$ when crossing the regions of bulk extended states, indicating that the topological nature of the extended states is indeed linked to the spin-Chern numbers. For large horizontal Zeeman fields, the spin mobility gap closes at strong disorder, prompting a change in the quantized spin-Chern numbers without a closing of the energy mobility gap.