  Many-body systems constructed of quantum-optical building blocks can now be
realized in experimental platforms ranging from exciton-polariton fluids to
ultracold gases of Rydberg atoms, establishing a fascinating interface between
traditional many-body physics and the driven-dissipative, non-equilibrium
setting of cavity-QED. At this interface, the standard techniques and
intuitions of both fields are called into question, obscuring issues as
fundamental as the role of fluctuations, dimensionality, and symmetry on the
nature of collective behavior and phase transitions. Here, we study the
driven-dissipative Bose-Hubbard model, a minimal description of numerous
atomic, optical, and solid-state systems in which particle loss is countered by
coherent driving. Despite being a lattice version of optical bistability---a
foundational and patently non-equilibrium model of cavity-QED---the steady
state possesses an emergent equilibrium description in terms of a classical
Ising model. We establish this picture by identifying a limit in which the
quantum dynamics is asymptotically equivalent to non-equilibrium Langevin
equations, which support a phase transition described by model A of the
Hohenberg-Halperin classification. Numerical simulations of the Langevin
equations corroborate this picture, producing results consistent with the
behavior of a finite-temperature Ising model.
