Non-Equilibrium Steady State generated by a moving defect: the supersonic threshold

We consider the dynamics of a system of free fermions on a 1D lattice in the presence of a defect moving at constant velocity. The defect has the form of a localized time-dependent variation of the chemical potential and induces at long times a Non-Equilibrium Steady State (NESS), which spreads around the defect. We present a general formulation which allows recasting the time-dependent protocol in a scattering problem on a static potential. We obtain a complete characterization of the NESS. In particular, we show a strong dependence on the defect velocity and the existence of a sharp threshold when such velocity exceeds the speed of sound. Beyond this value, the NESS is not produced and remarkably the defect travels without significantly perturbing the system. We present an exact solution for a $\delta-$like defect traveling with an arbitrary velocity and we develop a semiclassical approximation which provides accurate results for smooth defects.