Driven-dissipative ising model: Dynamical crossover at weak dissipation

Driven quantum systems coupled to an environment typically exhibit effectively thermal behavior with relaxational dynamics near criticality. However, a different qualitative might be expected in the weakly dissipative limit due competition between coherent and weak dissipation. In this work, we investigate driven-dissipative infinite-range Ising model presence of individual atomic dissipation, that emerges from paradigmatic open Dicke large-detuning limit. We show system undergoes dynamical crossover dynamics, characteristic exponent $\zeta=1/2$, underdamped critical governed by $\zeta=1/4$ regime; is markedly distinct equilibrium. Finally, utilizing exact diagrammatic representation, demonstrate criticality not artifact mean-field nature persists even short-range perturbations.