Fingerprint of band inversion in gating-dependent spin Hall conductance for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric topological insulators

It has always been a difficult issue to experimentally identify topological materials, though we can theoretically define and calculate various topological indices, such as the ${\mathbb{Z}}_{2}$ index. In this paper, we conduct first-principles calculations to investigate the spin Hall effect for topological insulators with $\mathcal{PT}$ symmetry and find that the gating dependence of the static SHC exhibits a five-stage evolution with band inversion and a three-stage evolution without band inversion, while the dynamical SHC features a reverse resonance peak in the frequency-dependent curve. We demonstrate the spin Hall conductivity with these signatures for three typical topological materials, for example, nonmagnetic topological insulator ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, magnetic static axion insulator ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$, and magnetic dynamical axion insulator ${\mathrm{Mn}}_{2}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{5}$. Our finding provides another perspective from the transportation signal to reveal the topological materials in experiments.