  Dipole-dipole interactions which govern phenomena like cooperative Lamb
shifts, superradiant decay rates, Van der Waals forces, as well as resonance
energy transfer rates are conventionally limited to the Coulombic near-field.
Here, we reveal a class of real-photon and virtual-photon long-range quantum
electrodynamic (QED) interactions that have a singularity in media with
hyperbolic dispersion. The singularity in the dipole-dipole coupling, referred
to as a Super-Coulombic interaction, is a result of an effective interaction
distance that goes to zero in the ideal limit irrespective of the physical
distance. We investigate the entire landscape of atom-atom interactions in
hyperbolic media and propose practical implementations with phonon-polaritonic
hexagonal boron nitride in the infrared spectral range and plasmonic
super-lattice structures in the visible range. Our work paves the way for the
control of cold atoms in hyperbolic media and the study of many-body atomic
states where optical phonons mediate quantum interactions.

Atomic Physics;Mesoscale and Nanoscale Physics;Materials Science;Optics