What is the nature of phase transitions in driven quantum systems? We address this question in two opposite regimes, a rapidly periodically driven Floquet system coupled to a bath and a slowly driven one, within a non-equilibrium renormalization group approach.

In the periodically driven case, the infinitely rapidly driven limit exhibits a second order phase transition analogous to a multi-mode laser. However, we reveal that fluctuations turn the transition first order when the driving frequency is finite. This can be traced back to a universal mechanism, which crucially hinges on the competition of degenerate, near critical modes associated to higher Floquet Brillouin zones, in some analogy to the Halperin-Lubensky-Ma or Coleman-Weinberg mechanism. A physical picture is that of a fluctuation induced many-body Kapitza pendulum. The critical exponents of the infinitely rapidly driven system -- including a new, independent one -- can yet be probed experimentally upon smoothly tuning towards that limit.

The opposite, slowly driven limit, is governed by the Kibble-Zurek mechanism of diabatic freeze out of the divergence of the correlation length. Recasting this problem in a systematic RG formulation, we show that the slow drive can be used to resolve not only the leading critical exponents of the underlying equilibrium problem, but the full critical exponent spectrum.