  Quantum simulators employing cold atoms are among the most promising
approaches to tackle quantum many-body problems. Nanophotonic structures are
widely employed to engineer the bandstructure of light and are thus
investigated as a means to tune the interactions between atoms placed in their
vicinity. A key shortcoming of this approach is that excitations can decay into
free photons, limiting the coherence of such quantum simulators. Here, we
overcome this challenge by proposing to use a simple cubic three-dimensional
array of atoms to produce an omnidirectional bandgap for light and show that it
enables coherent, dissipation-free interactions between embedded impurities. We
show explicitly that the band gaps persist for moderate lattice sizes and
finite filling fraction, which makes this effect readily observable in
experiment. Our work paves the way toward analogue spin quantum simulators with
long-range interactions using ultracold atomic lattices, and is an instance of
the emerging field of atomic quantum metamaterials.
