The past decade has seen extensive work on building a systematic Wilsonian effective field theory for dissipative and stochastic dynamics in systems in thermal equilibrium: in particular, an effective field theory of hydrodynamics. In this talk, I will argue that the same principles that underlie this construction can also be applied to certain systems that are not in equilibrium. In particular, I will discuss the classification of generic stochastic dynamics that drive a few- or many-body system towards a target statistical steady state. Such models come with a natural notion of time-reversal symmetry, which can be broken in controlled ways, leading to nonequilibrium dynamical universality classes. I will overview the application of our approach in two many-body settings: (1) “odd elasticity”, which generalizes the conventional theory of solid mechanics to active systems; (2) a cartoon model of “failed flocking” — an O(2) rotor model on the square lattice where spins appear to “flock” in the direction they are pointing, and yet no long-range order can arise, in contrast to prior field-theoretic expectations. Our methods can lead to a more systematic classification of nonequilibrium dynamical universality classes and a more detailed understanding of (emergent) symmetries in nonequilibrium effective theories.
