Tunable lower critical fractal dimension for a non-equilibrium phase transition

We theoretically investigate the role of spatial dimension and driving frequency in a non-equilibrium phase transition of a driven-dissipative interacting bosonic system. In this setting, spatial dimension is dictated by the shape of the external driving field. We consider both homogeneous driving configurations, which correspond to standard integer-dimensional systems, and fractal driving patterns, which give rise to a non-integer Hausdorff dimension for the spatial density. The onset of criticality is characterized by critical slowing down in the excited density dynamics as the system asymptotically approaches the steady state. By analyzing the system-size dependence of the asymptotic decay rate using numerical simulations of the full multi-mode dynamics, complemented by an analytical statistical mean-field treatment, we determine the lower critical dimension of the non-equilibrium phase transition. We show that this dimension can be non-integer and fractal in nature, and that it can be tuned continuously via the frequency detuning of the driving field.