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We present a comprehensive analysis of critical behavior in thedriven-dissipative Bose condensation transition in three spatial dimensions.Starting point is a microscopic description of the system in terms of amany-body quantum master equation, where coherent and driven-dissipativedynamics occur on an equal footing. An equivalent Keldysh real time functionalintegral reformulation opens up the problem to a practical evaluation using thetools of quantum field theory. In particular, we develop a functionalrenormalization group approach to quantitatively explore the universality classof this stationary non-equilibrium system. Key results comprise the emergenceof an asymptotic thermalization of the distribution function, while manifestnon-equilibrium properties are witnessed in the response properties in terms ofa new, independent critical exponent. Thus the driven-dissipative microscopicnature is seen to bear observable consequences on the largest length scales.The absence of two symmetries present in closed equilibrium systems -underlying particle number conservation and detailed balance, respectively - isidentified as the root of this new non-equilibrium critical behavior. Ourresults are relevant for broad ranges of open quantum systems on the interfaceof quantum optics and many-body physics, from exciton-polariton condensates tocold atomic gases.

Non-equilibrium Functional Renormalization for Driven-Dissipative Bose-Einstein Condensation