Rényi entropies of generic thermodynamic macrostates in integrable systems

We study the behaviour of Rényi entropies in a generic thermodynamic macrostate of an integrable model. In the standard quench action approach to quench dynamics, the Rényi entropies may be derived from the overlaps of the initial state with Bethe eigenstates. These overlaps fix the driving term in the thermodynamic Bethe ansatz (TBA) formalism. We show that this driving term can be also reconstructed starting from the macrostate's particle densities. We then compute explicitly the stationary Rényi entropies after the quench from the dimer and the tilted Néel state in XXZ spin chains. For the former state we employ the overlap TBA approach, while for the latter we reconstruct the driving terms from the macrostate. We discuss in full detail the limits that can be analytically handled and we use numerical simulations to check our results against the large time limit of the entanglement entropies.