RESEARCH ARTICLE Superconductor-normal metal quantum phase transition in dissipative and non-equilibrium systems

In physical systems, coupling to the environment gives rise to dissipation and decoherence. For nanoscopic materials, this may be a determining factor of their physical behaviour. However, even for macroscopic many-body systems, if the strength of this coupling is sufficiently strong, their ground-state properties and phase diagram may be severely modified.Also dissipation is essential to allow a system in the presence of a time-dependent perturbation to attain a steady, timeindependentstate.Inthiscase,thenon-equilibriumphasediagramdependsonthe intensity of the perturbation and on the strength of the coupling of the system to the outside world. In this paper, we investigate the effects of both dissipation and time-dependent external sources in the phase diagram of a many-body system at zero and finite temperatures. For concreteness, we consider the specific case of a superconducting layer under the action of an electric field and coupled to a metallic substrate. The former arises from a time dependent vector potential minimally coupled to the electrons in the layer.We introduce a Keldysh approach that allows to obtain the time dependence of the superconducting order parameter in an adiabatic regime. We study the phase diagram of this system as a function of the electric field, the coupling to the metallic substrate and temperature.