A non-perturbative no-go theorem for photon condensation in approximate models

Equilibrium phase transitions between a normal and a photon condensate state (also known as super-radiant 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 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. Here, we demonstrate a general no-go theorem valid also for truncated, gauge-invariant models which forbids first-order as well as second-order super-radiant 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.