  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. Recent 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 {\it spatially uniform}.
However, 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. %{It has also been recently shown that truncation of the
Hilbert space of the matter system can affect the gauge invariance of the
theory. However, it is always possible to obtain approximate Hamiltonians
obeying the gauge principle in the truncated Hilbert space.} Here, we
demonstrate a general no-go theorem valid also for truncated, gauge-invariant
models which forbid first-order as well as second-order superradiant phase
transitions in the absence of a coupling with a magnetic field. In particular,
we explicitly consider the cases of interacting electrons in a lattice and
$M$-level systems.
