  Recent theories predict dissipationless spin current induced by an electric
field in doped semiconductors. Nevertheless, the charge current is still
dissipative in these systems. In this work, we theoretically predict the
dissipationless spin Hall effect, without any accompanying charge current, in
some classes of band insulators, including zero-gap semiconductors such as HgTe
and narrow-gap semiconductors such as PbTe. This effect is similar to the
quantum Hall effect in that all the states below the gap contribute and there
occurs no dissipation. However the spin Hall conductance is not quantized even
in two dimensions. This is the first example of a nontrivial topological
structure in a band insulator without any magnetic field.
