Symmetry breaking in open systems and the onset of semi-classical dynamics

The problem of how many-body systems approach equilibrium has a long history. In the study of dynamical critical phenomena, such dynamics is modeled with the Langevin equation of order parameter fields, and a classification and universal description can be obtained given the symmetry properties of the fields. With the developments of quantum platforms and the need to reliably prepare Gibbs states on such platforms, it is timely to revisit these results from an open quantum system perspective. In this work, we study Lindbladian dynamics targeting various Gibbs states. The notion of symmetry needs to be enriched to include both strong symmetries and weak ones. We start by constructing explicit Lindbladians that target Gibbs ensembles described by the same classical field theory appearing in dynamical critical phenomena, with the classical symmetries promoted to strong symmetries on the Lindbladian. When our dynamics have a strong Z2 symmetry, we show that the late-time dynamics is accompanied by either a multiplicative Langevin noise when the strong symmetry is not spontaneously broken, or reproduces known model A dynamics with an additive noise when the symmetry spontaneously breaks. We also study dynamics with a strong U(1) symmetry and extract the time scale associated with the onset of semi-classical dissipation. Exactly solvable models and numerical evidence are presented to quantitatively establish these time scales.