  The structure of solids and their phases is mainly determined by static
Coulomb forces while the coupling of charges to the dynamical, i.e., quantized
degrees of freedom of the electromagnetic field plays only a secondary role.
Recently, it has been speculated that this general rule can be overcome in the
context of cavity quantum electrodynamics (QED), where the coupling of dipoles
to a single field mode can be dramatically enhanced. Here we present a first
exact analysis of the ground states of a dipolar cavity QED system in the
non-perturbative coupling regime, where electrostatic and dynamical
interactions play an equally important role. Specifically, we show how strong
and long-range vacuum fluctuations modify the states of dipolar matter and
induce novel phases with unusual properties. Beyond a purely fundamental
interest, these general mechanisms can be important for potential applications,
ranging from cavity-assisted chemistry to quantum technologies based on
ultrastrongly coupled circuit QED systems.
