  We analyze a quantum walk on a bipartite one-dimensional lattice, in which
the particle can decay whenever it visits one of the two sublattices. The
corresponding non-Hermitian tight-binding problem with a complex potential for
the decaying sites exhibits two different phases, distinguished by a winding
number defined in terms of the Bloch eigenstates in the Brillouin zone. We find
that the mean displacement of a particle initially localized on one of the
non-decaying sites can be expressed in terms of the winding number, and is
therefore quantized as an integer, changing from zero to one at the critical
point. This problem can serve as a simplified model for nuclear spin pumping in
the spin-blockaded electron transport regime of quantum dots in the presence of
competing hyperfine and spin-orbital interactions. The predicted transition
from pumping to non-pumping is topological in nature, and is hence robust
against certain types of noise and decoherence.
