Quench dynamics of noninteracting fermions with a delta impurity

We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength $g$ is varied at time $t=0$. The system is prepared in its ground state with $g=g_0=+\infty$, with two different densities and Fermi wave-vectors $k_L$ and $k_R$ on the two half-spaces $x>0$ and $x<0$ respectively. It then evolves for $t>0$ as an isolated system, with a finite impurity strength $g$. We compute exactly the time dependent density and current. For a fixed position $x$ and in the large time limit $t \to \infty$, the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of "rays", $x/t=\xi$ fixed with $x, t \to \infty$, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities $k_L$ and $k_R$ in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities $\xi$ and $-\xi$ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.