We show that non-Hermiticity enables topological phases with unidirectional transport in one-dimensional Floquet chains. The topological signatures of these phases are non-contractible loops in the spectrum of the Floquet propagator that are separated by an imaginary gap. Such loops occur exclusively in non-Hermitian Floquet systems. We define the corresponding topological invariant as the winding number of the Floquet propagator relative to the imaginary gap. To relate topology to transport, we introduce the concept of regularized dynamics of non-Hermitian chains. We establish that, under the conditions of regularized dynamics, transport is quantized in so far as the charge transferred over one period equals the topological winding number. We illustrate these theoretical findings with the example of a Floquet chain that features a topological phase transition and acts as a charge pump in the non-trivial topological phase. We finally discuss whether these findings justify the notion that non-Hermitian Floquet chains support topological transport.
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