Abstract
Following our previous work S. Murakami, N. Nagaosa, S. C. Zhang, Science
301, 1348 (2003) on the dissipationless quantum spin current, we present an
exact quantum mechanical calculation of this novel effect based on the linear
response theory and the Kubo formula. We show that it is possibxle to define an
exactly conserved spin current, even in the presence of the spin-orbit coupling
in the Luttinger Hamiltonian of p-type semiconductors. The light- and the
heavy-hole bands form two Kramers doublets, and an SU(2) non-abelian gauge
field acts naturally on each of the doublets. This quantum holonomy gives rise
to a monopole structure in momentum space, whose curvature tensor directly
leads to the novel dissipationless spin Hall effect, i.e., a transverse spin
current is generated by an electric field. The result obtained in the current
work gives a quantum correction to the spin current obtained in the previous
semiclassical approximation.
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