  A potential phase transition between a normal ground state and a
photon-condensed ground state in many-dipole light-matter systems is a topic of
considerable controversy, exasperated by conflicting no-go and counter no-go
theorems and often ill-defined models. We clarify this long-lasting debate by
analyzing two specific arrangements of atoms, including a 3D cubic lattice and
a cavity-embedded square lattice layer, which provides a physical model for
single-mode cavity QED with coupled dipoles in the thermodynamic limit. These
models are shown to significantly differ from the standard Dicke model and, in
the thermodynamic limit, give rise to renormalized Hopfield models. We show
that a ferroelectric phase transition can (in principle) still occur and the
description of the abnormal phase beyond the critical point requires the
inclusion of nonlinear terms in the Holstein-Primakoff mapping. We also show
how our model agrees with recent experiments.
