  We introduce and analyze a class of one dimensional insulating Hamiltonians
which, when adiabatically varied in an appropriate closed cycle, define a
"$Z_2$ pump". For an isolated system a single closed cycle of the pump changes
the expectation value of the spin at each end even when spin orbit interactions
violate the conservation of spin. A second cycle, however returns the system to
its original state. When coupled to leads, we show that the $Z_2$ pump
functions as a spin pump in a sense which we define, and transmits a finite,
though non quantized spin in each cycle. We show that the $Z_2$ pump is
characterized by a $Z_2$ topological invariant that is analogous to the Chern
invariant that characterizes a topological charge pump. The $Z_2$ pump is
closely related to the quantum spin Hall effect, which is characterized by a
related $Z_2$ invariant. This work presents an alternative formulation which
clarifies both the physical and mathematical meaning of that invariant. A
crucial role is played by time reversal symmetry, and we introduce the concept
of the time reversal polarization, which characterizes time reversal invariant
Hamiltonians and signals the presence or absence of Kramers degenerate end
states. For non interacting electrons we derive a formula for the time reversal
polarization which is analogous to the Berry's phase formulation of the charge
polarization. For interacting electrons, we show that abelian bosonization
provides a simple formulation of the time reversal polarization. We discuss
implications for the quantum spin Hall effect, and argue in particular that the
$Z_2$ classification of the quantum spin Hall effect is valid in the presence
of electron electron interactions.
