  Non-Hermitian quantum systems can exhibit unique observables characterizing
topologically protected transport in the presence of decay. The topological
protection arises from winding numbers associated with non-decaying dark
states, which are decoupled from the environment and thus immune to
dissipation. Here we develop a classification of topological dynamical phases
for one-dimensional quantum systems with periodically-arranged absorbing sites.
This is done using the framework of Bloch theory to describe the dark states
and associated topological invariants. The observables, such as the average
particle displacement over its life span, feature quantized contributions that
are governed by the winding numbers of cycles around dark-state submanifolds in
the Hamiltonian parameter space. Changes in the winding numbers at topological
transitions are manifested in non-analytic behavior of the observables. We
discuss the conditions under which nontrivial topological phases may be found,
and provide examples that demonstrate how additional constraints or symmetries
can lead to rich topological phase diagrams.
