Dynamical free energy and the Loschmidt-echo for a class of quantum quenches in the Heisenberg spin chain

We consider a class of global quantum quenches in the Heisenberg XXZ spin chain, where the initial states are given by products of local two-site states. The two main examples are the N\'eel state and the dimer state. We derive an exact analytic result for the "Loschmidt echo per site" at imaginary times and also consider the analytic continuation back to real times. As a by-product we obtain an exact result for the "overlap per site" between the N\'eel state and the ground state of the XXZ Hamiltonian in the massive regime.