Dynamical phase transitions in two-dimensional Brownian matter

We investigate collective behavior of a system of two-dimensional interacting Brownian particles in the hydrodynamic regime. By means of the Martin–Siggia–Rose–Jenssen–de Dominicis formalism, we built up a generating functional for correlations functions. In the continuum limit, we uncover an exact symmetry under area-preserving diffeomorphism transformations that characterizes a liquid state. This symmetry leads to the conservation of local vorticity. By computing the generating functional within the saddle-point plus Gaussian fluctuations approximation, we reveal the emergence of a $U\left(1\right)$ gauge symmetry that allows us to describe the dynamics of density fluctuations as a gauge theory. We solve the corresponding equations of motion for short as well as long ranged interactions showing up the presence of multiple dynamical regimes and associated dynamical phase transitions, even for pure repulsive interactions.