  Equilibrium phase transitions between a normal and a photon condensate state
(also known as superradiant phase transitions) are a highly debated research
topic, where proposals for their occurrence and no-go theorems have chased each
other for the past four decades. Previous no-go theorems have demonstrated that
gauge invariance forbids second-order phase transitions to a photon condensate
state when the cavity-photon mode is assumed to be spatially uniform.
First-order phase transitions were previously considered as possible paths to
bypass the no-go theorem. In particular, it has been theoretically predicted
that a collection of three-level systems coupled to light can display a
first-order phase transition to a photon condensate state. Here, we lay down a
general no-go theorem, which forbids first-order as well as second-order
superradiant phase transitions in a spatially-uniform cavity field. By using
the tools of lattice gauge theory, we then derive a fully gauge-invariant model
for three-level systems coupled to light. In agreement with the general
theorem, our model does not display photon condensation.
