Driving and dissipation can stabilize Bose-Einstein condensates. Using Keldysh field theory, we analyze this phenomenon for Markovian systems that can comprise on-site two-particle driving, on-site single-particle and two-particle loss, as well as edge-correlated pumping. Above the upper critical dimension, mean-field theory shows that pumping and two-particle driving induce condensation right at the boundary between the stable and unstable regions of the noninteracting theory. With nonzero two-particle driving, the condensate is gapped. This picture is consistent with the recent observation that, without symmetry constraints beyond invariance under single-particle basis transformations, all gapped quadratic bosonic Liouvillians belong to the same phase. For systems below the upper critical dimension, the edge-correlated pumping penalizes high-momentum fluctuations, rendering the theory renormalizable. We perform the one-loop renormalization group analysis, finding a condensation transition inside the unstable region of the noninteracting theory. Interestingly, its critical behavior is determined by a Wilson-Fisher-like fixed point with universal correlation-length exponent $\nu =0.6$ in three dimensions.

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• Collective effects in quantum optics
• Critical exponents
• Critical phenomena
• Dynamic critical phenomena
• Open quantum systems
• Open quantum systems & decoherence
• Bose gases
• Bose-Einstein condensates
• Nonequilibrium systems
• Ultracold gases
• Lindblad equation
• Nonequilibrium Green's function