Time reversal polarization and a Z 2 adiabatic spin pump

We introduce and analyze a class of one-dimensional insulating Hamiltonians that, 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 that we define, and transmits a finite, though nonquantized, 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…