  We study a driven-dissipative model of spins one-half (qubits) on a lattice
with nearest-neighbor interactions. Focusing on the role of spatially extended
spin-spin correlations in determining the phases of the system, we characterize
the spatial structure of the correlations in the steady state, as well as their
temporal dynamics. In dimension one we use essentially exact
matrix-product-operator simulations on large systems, and pushing these
calculations to dimension two, we obtain accurate results on small cylinders.
We also employ an approximation scheme based on solving the dynamics of the
mean field dressed by the feedback of quantum fluctuations at leading order.
This approach allows us to study the effect of correlations in large lattices
with over one hundred thousand spins, as the spatial dimension is increased up
to five. In dimension two and higher we find two new states that are stabilized
by quantum correlations and do not exist in the mean-field limit of the model.
One of these is a steady state with mean magnetization values that lie between
the two bistable mean-field values, and whose correlation functions have
properties reminiscent of both. The correlation length of the new phase
diverges at a critical point, beyond which we find emerging a new limit cycle
state with the magnetization and correlators oscillating periodically in time.
