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

We study the Renyi entropies in the spin-1/2 anisotropic Heisenberg chain after a quantum quench starting from the Neel state. The quench action method allows us to obtain the stationary Renyi entropies for arbitrary values of the index alpha as generalised free energies evaluated over a calculable thermodynamic macrostate depending on alpha. We work out this macrostate for several values of alpha and of the anisotropy Delta by solving the thermodynamic Bethe ansatz equations. By varying alpha different regions of the Hamiltonian spectrum are accessed. The two extremes are alpha -> infinity for which the thermodynamic macrostate is either the ground state or a low-lying excited state ( depending on Delta) and alpha = 0 when the macrostate is the infinite temperature state. The Renyi entropies are easily obtained from the macrostate as function of alpha 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 Renyi entanglement entropies. For large subsystems and for any alpha, their density turns out to be compatible with that of the thermodynamic Renyi entropies.