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

TL;DR

This study demonstrates that the spin Hall conductivity in 2D topological insulators is quantized under certain symmetries and is robust against various perturbations, with deviations linked to symmetry and spin-orbit effects.

Contribution

It provides a detailed numerical analysis of the quantization of spin Hall conductivity in 2D topological insulators considering symmetry and spin-orbit interaction effects.

Findings

Quantized spin Hall conductivity observed in highly symmetric 2D insulators.

Rectangular symmetry systems show significantly reduced conductivity.

Symmetry-breaking fields do not affect the quantization in the topological phase.

Abstract

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