Network Model for $Z_2$ Quantum Spin-Hall Effects with Disorder

We study the effects of static disorder on the $Z_2$ quantum spin-Hall effect for noninteracting electrons propagating in two dimensions. To this end, a network model realizing the $Z_2$ quantum spin-Hall effect is constructed to account for the effects of static disorder on the propagation of noninteracting electrons subjected to spin-orbit couplings with the time-reversal symmetry. This network model is different from past network models belonging to the symplectic symmetry class in that the propagating modes along the links of the network has a single Kramers doublet. By investigating this network model numerically, it is found that a two-dimensional metallic phase of finite extent is embedded in insulating phases. We also find that the Anderson localization-delocalization transition between the metallic and $Z_2$ insulating phases belong to the conventional symplectic universality class in two dimensions.