Driven-dissipative quantum systems have become an important paradigm of nonequilibrium quantum systems, and aptly describe many common experimental setups in atomic, molecular, and optical physics. However, they are not understood as well as their equilibrium counterparts, and analytical solutions to the nonequilibrium dynamics are few and far between. Furthermore, phase transitions in driven-dissipative systems may host new universality classes, or connect to universality classes seen in equilibrium. In this work, we present a thorough analytical and numerical treatment of the driven-dissipative Ising model with infinite-range interactions and local spontaneous emission. This model is amenable to an exact field-theoretical solution via a quantum-to-classical mapping. We primarily focus on the critical properties, which show interesting similarities and differences in comparison with equilibrium and classical phase transitions. Notably, we identify two distinct universality classes in the phase diagram. A generic point on along the phase boundary falls under the same universality class as the infinite range classical Ising model with Glauber dynamics. However, in the weakly-dissipative limit we find that the system is in the same universality class as the finite-temperature equilibrium transition of the quantum Ising model. Furthermore, we discover a new notion of time-reversal symmetry which occurs near the phase boundary due to the interplay of drive and dissipation. Finally, we characterize various measures of entanglement in the model throughout the phase diagram by calculating the quantites known as the quantum Fisher information, the logarithmic negativity, and spin squeezing. We complement these findings by also calculating measures of total correlations in the system such as the von Neumann entropy and the mutual information.
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