Renyi entropies after releasing the Neel state in the XXZ spin-chain

We study the R\'enyi entropies in the spin-1/2 anisotropic Heisenberg chain after a quantum quench starting from the N\'eel state. The quench action method allows us to obtain the stationary R\'enyi entropies for arbitrary values of the index \u3b1 as generalised free energies evaluated over a calculable thermodynamic macrostate depending on \u3b1. We work out this macrostate for several values of \u3b1 and of the anisotropy \u394 by solving the thermodynamic Bethe ansatz equations. By varying \u3b1 different regions of the Hamiltonian spectrum are accessed. The two extremes are \u3b1\u2192 1e for which the thermodynamic macrostate is either the ground state or a low-lying excited state (depending on \u394) and \u3b1=0 when the macrostate is the infinite temperature state. The R\'enyi entropies are easily obtained from the macrostate as function of \u3b1 and a few interesting limits are analytically characterised. We provide robust numerical evidence to confirm our results using exact diagonalisation and a stochastic numerical implementation of Bethe ansatz. Finally, using tDMRG we calculate the time evolution of the R\'enyi entanglement entropies. For large subsystems and for any \u3b1, their density turns out to be compatible with that of the thermodynamic R\'enyi entropie