  Topological phases in quantum and classical systems have been of significant
recent interest due to their fascinating physical properties. While a range of
different mechanisms to induce topological order have been introduced, a quest
for the most viable solution for practical systems is still open. Floquet
topological insulator represent one of possible venues to engineer topological
phases, yet they have been so far largely restricted to temporal modulation of
Hermitian potentials. On the other hand, in many physical systems, including
acoustic and optical systems, modulating loss or gain can be more
straightforwardly achieved. Two of such examples are graphene, which enables
strong modulation of its conductivity due to saturable absorption, and quantum
wells where population inversion can be achieved in an ultrafast manner. On the
other hand, non-Hermitian Floquet potentials have not been shown to yield novel
topological phases to date. It is therefore of great interest to explore
time-modulated non-Hermitian potentials in periodic lattices, and the emergence
of topological phases associated with them. Here we demonstrate that
non-Hermitian Hamiltonians can indeed result in topological phases supporting
nonreciprocal edge states propagating without dissipation, as well as new
regimes of dissipative and amplifying topological edge transport.
