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Inhomogeneous quenches in a 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 the same or different. We present a number of exact results for the correlation functions in the non-interacting case. We present a short-time expansion, which can sometimes be fully resummed, and which reproduces the so-called `light cone' effect or wavefront behavior of the correlators. For large times, we show how all interesting physical regimes may be obtained by stationary phase approximation techniques. In particular, we derive semiclassical formulas in the case when both time and positions are large, and show that these are exact in the thermodynamic limit. We present subleading corrections to the large-time behavior, including the corrections near the edges of the wavefront. We also provide results for the return probability or Loschmidt echo. In the maximally inhomogeneous limit, we prove that it is exactly gaussian at all times. The effects of interactions on the Loschmidt echo are also discussed.