Inhomogeneous quenches in a free fermionic chain: Exact results

We consider the non-equilibrium physics induced by joining together two tight-binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be different. We present a number of exact results, focusing on two-point correlators and the Loschmidt echo (return probability). For the non-interacting case, we identify through an exact derivation the regime in which a semiclassical ansatz is valid. We present a number of analytical results beyond semiclassics, such as the approach to the non-equilibrium steady state and the appearance of Tracy-Widom distributions at the front of the light cone. The light cone behavior is quantified through a series expansion in time, and this description is shown to be valid for interacting systems as well. Results on the Loschmidt echo, presented for finite and zero interactions, illustrate that the physics is different from both local and global quenches