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The third-rank tensor of the static spin Hall conductivity is investigatedfor two-dimensional (2D) topological insulators by electronic structurecalculations. Its seeming quantization is numerically demonstrated for highlysymmetric systems independent of the gap size. 2D crystals with hexagonal andsquare Bravais lattice show similar effects, while true rectangulartranslational symmetry yields conductivity values much below the quantum$e^2/h$. Field-induced lifting the inversion symmetry does not influence thequantum spin Hall state up to band inversion but the conductivity quantization.Weak symmetry-conserving biaxial but also uniaxial strain has a minor influenceas long as inverted gaps dictate the topological character. The results arediscussed in terms of the atomic geometry and the Rashba contribution to thespin-orbit interaction (SOI). Translational and point-group symmetry as well asSOI rule the deviation from the quantization of the spin Hall conductance.

Quantization of spin Hall conductivity in two-dimensional topological insulators versus symmetry and spin-orbit interaction