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We introduce a method to evaluate the steady-state non-equilibriumKeldysh-Schwinger Green's functions for infinite systems subject to both anelectric field and a coupling to reservoirs. The method we present exploits aphysical quasi-translation invariance, where a shift by one unit cell leavesthe physics invariant if all electronic energies are simultaneously shifted bythe magnitude of the electric field. Our framework is straightaway applicableto diagrammatic many-body methods. We discuss two flagship applications,mean-field theories as well as a sophisticated second-order functionalrenormalization group approach. The latter allows us to push therenormalization-group characterization of phase transitions for latticefermions into the out-of-equilibrium realm. We exemplify this by studying amodel of spinless fermions, which in equilibrium exhibits aBerezinskii-Kosterlitz-Thouless phase transition.

Phases of translation-invariant systems out of equilibrium: Iterative Green's function techniques and renormalization group approaches