Driving quantum materials with coherent light has proven a powerful platform to realize a plethora of interesting phases and transitions, ranging from ferroelectricity to superconductivity and limit cycles in pumped magnonics. In this paper we develop the field theoretical framework to describe nonequilibrium phases that emerge in systems pumped by rapid parametric drives. We consider paradigmatic $\text{O}\left(N\right)$ models that describe the long-wavelength fluctuations of ordering fields in many condensed matter setups. We show that rapid parametric driving of these models can induce an effective pump mechanism in the long-wavelength regime through nonlinear scattering. This induces a nonequilibrium transition into a time-crystalline phase.

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• Nonequilibrium systems
• Langevin equation
• Path-integral methods