Fragile fate of driven-dissipative XY phase in two dimensions

Driven-dissipative systems define a broad class of nonequilibrium systems where an external drive (e.g., laser) competes with a dissipative environment. The steady state of dynamics is generically distinct from a thermal state characteristic of equilibrium. As a representative example, a driven-dissipative system with a continuous symmetry is generically disordered in two dimensions in contrast with the well-known algebraic order in equilibrium XY phases. In this paper, we study a two-dimensional driven-dissipative model of weakly interacting bosons with a continuous $U\left(1\right)$ symmetry. Our aim is twofold: First, we show that an effectively equilibrium XY phase emerges despite the driven nature of the model, and that it is protected by a natural ${\mathbb{Z}}_2$ symmetry of the dynamics. Second, we argue that this phase is unstable against symmetry-breaking perturbations as well as static disorder, whose mechanism in most cases has no analog in equilibrium. In the language of renormalization group theory, we find that, outside equilibrium, there are more relevant directions away from the XY phase.

Figure 1

Emergence of an effectively thermal XY phase as the steady state of a nonequilibrium driven-dissipative model with ${\mathbb{Z}}_2\times U\left(1\right)$ symmetry of the unit cell. The schematic plot shows relevant perturbations away from the XY phase, which include symmetry-breaking perturbations as well as static disorder. While $U\left(1\right)$ symmetry breaking finds an effective equilibrium character, the corresponding mechanisms for ${\mathbb{Z}}_2$ symmetry breaking and disorder are of a genuinely nonequilibrium nature. (The highlighted plane represents the subspace spanned by genuinely nonequilibrium perturbations.)