  Giant atoms -- quantum emitters that couple to light at multiple discrete
points -- are emerging as a new paradigm in quantum optics thanks to their many
promising properties, such as decoherence-free interaction. While most previous
work has considered giant atoms coupled to open continuous waveguides or a
single giant atom coupled to a structured bath, here we study the interaction
between two giant atoms mediated by a structured waveguide, e.g., a photonic
crystal waveguide. This environment is characterized by a finite energy band
and a band gap, which affect atomic dynamics beyond the Markovian regime. Here
we show that, inside the band, decoherence-free interaction is possible for
different atom-cavity detunings, but is degraded from the continuous-waveguide
case by time delay and other non-Markovian effects. Outside the band, where
atoms interact through the overlap of bound states, we find that giant atoms
can interact more strongly and over longer distances than small atoms for some
parameters -- for instance, when restricting the maximum coupling strength
achievable per coupling point. The results presented here may find applications
in quantum simulation and quantum gate implementation.
