Exact time evolution formulae in the XXZ spin chain with domain wall initial state

We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but $k$ spins come back to their initial orientation, as a $k-$fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.