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We develop a quantum dynamical field theory for studying phase transitions indriven open systems coupled to Markovian noise, where non-linear noise effectsand fluctuations beyond semiclassical approximations influence the criticalbehaviour. We systematically compare the diagrammatics, the properties of therenormalization group flow and the structure of the fixed points, of the novelquantum dynamical field theory and of its semi-classical counterpart, which isemployed to characterise dynamical criticality in three dimensionaldriven-dissipative condensates. As an application, we perform the KeldyshFunctional Renormalization of a one dimensional driven open Bose gas, where atailored diffusion Markov noise realises an analog of quantum criticality fordriven-dissipative condensation. We find that the associated non-equilibriumquantum phase transition does not map into the critical behaviour of its threedimensional classical driven counterpart.

Quantum dynamical field theory for non-equilibrium phase transitions in driven open systems