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The spin conductance of two-dimensional topological insulators (2D TIs) isnot expected to be quantized in the presence of perturbations that break thespin-rotational symmetry. However, the deviation from the pristine-limitquantization has yet to be studied in detail. In this paper, we define the spincurrent operator for the helical edge modes of a 2D TI and introduce afour-terminal setup to measure spin conductances. Using the developedformalism, we consider the effects of disorder terms that break spin-rotationalsymmetry or give rise to edge-to-edge coupling. We identify a key role playedby spin torque in an out-of-equilibrium edge. We then utilize a tight-bindingmodel of topological monolayer WTe$_2$ and scattering matrix formalism tonumerically study spin transport in a four-terminal 2D TI device. Inparticular, we calculate the spin conductances and characteristic spin decaylength in the presence of magnetic disorder. In addition, we study the effectsof inter-edge scattering in a quantum point contact geometry. We find that thespin Hall conductance is surprisingly robust to spin symmetry-breakingperturbations, as long as time-reversal symmetry is preserved and inter-edgescattering is weak.

Edge spin transport in the disordered two-dimensional topological insulator WTe$_2$