Approach to a stationary state in a driven Hubbard model coupled to a thermostat

We investigate the dynamics of the Hubbard model in a static electric field in order to identify the conditions necessary to reach a nonequilibrium stationary state. We show that, for a generic electric field, the convergence to a stationary state requires coupling to a thermostatting bath that absorbs the work done by the external field. Following the real-time dynamics of the system, we show that a nonequilibrium stationary state is reached for essentially any value of the coupling to the bath. We characterize the properties of such nonequilibrium stationary states by studying suitable physical observables, pointing out the existence of an analog of the Pomeranchuk effect as a function of the electric field. We map out a phase diagram in terms of dissipation and electric field strengths and identify the dissipation values at which the steady current is largest for a given field.