Cancellation of intrinsic ``spin-Hall'' conductivity in absence of broken time-reversal symmetry

A Streda-type argument is used to obtain the intrinsic dissipationless ``spin-Hall'' conductivity of electronic systems with spin-orbit coupling (SOC). Instead of directly calculating the ``spin-current'' response to an electric field, we calculate the {\it spin-density} response to a magnetic flux density $B$ (from orbital, rather than Zeeman coupling), and transform to a moving frame in which an electric field is present. This is consistent when used to compute the ``anomalous'' and ``quantized'' (electrical) Hall conductivities, and appears to also be so for the ``spin-Hall'' conductivity: the induced ``spin-current'' in the moving frame is interpreted as the induced spin density in the static frame, times the boost velocity. In the 2D model with ``Rashba'' spin-orbit coupling, the spin density induced by linear response to $B$ is cancelled by an ``anomalous'' term from the lowest Landau level, which is ``special'' because it alone is spin-unpaired. Similar unpaired ``special'' Landau levels also occur in the 3D ``Luttinger'' model for SOC of holes in p-type semiconductors. We also argue that a {\it quantized} spin-Hall effect cannot occur in the absence of broken time-reversal symmetry, and conjecture that this is also true for the metallic (non-quantized) spin-Hall effect.