  The driven, dissipative Bose-Hubbard model (BHM) provides a generic
description of collective phases of interacting photons in cavity arrays. In
the limit of strong optical nonlinearities (hard-core limit), the BHM maps on
the dissipative, transverse-field XY model (XYM). The steady-state of the XYM
can be analyzed using mean-field theory, which reveals a plethora of
interesting dynamical phenomena. For example, strong hopping combined with a
blue-detuned drive, leads to an instability of the homogeneous steady-state
with respect to antiferromagnetic fluctuations. In this paper, we address the
question whether such an antiferromagnetic instability survives in the presence
of quantum correlations beyond the mean-field approximation. For that purpose,
we employ a self-consistent $1/z$ expansion for the density matrix, where $z$
is the lattice coordination number, i.e., the number of nearest neighbours for
each site. We show that quantum fluctuations stabilize a new homogeneous
steady-state with antiferromagnetic correlations in agreement with exact
numerical simulations for finite lattices. The latter manifests itself as
short-ranged oscillations of the first and second-order spatial coherence
functions of the photons emitted by the array.

Quantum Gases