Quench action and Renyi entropies in integrable systems

Entropy is a fundamental concept in equilibrium statistical mechanics, yet its origin in the non-equilibrium dynamics of isolated quantum systems is not fully understood. A strong consensus is emerging around the idea that the stationary thermodynamic entropy is the von Neumann entanglement entropy of a large subsystem embedded in an infinite system. Also motivated by cold-atom experiments, here we consider the generalisation to Renyi entropies. We develop a new technique to calculate the diagonal Renyi entropy in the quench action formalism. In the spirit of the replica treatment for the entanglement entropy, the diagonal Renyi entropies are generalised free energies evaluated over a thermodynamic macrostate which depends on the Renyi index and, in particular, it is not the same describing the von Neumann entropy. The technical reason for this, maybe surprising, result is that the evaluation of the moments of the diagonal density matrix shifts the saddle point of the quench action. An interesting consequence is that different Renyi entropies encode information about different regions of the spectrum of the post-quench Hamiltonian. Our approach provides a very simple proof of the long-standing issue that, for integrable systems, the diagonal entropy is half of the thermodynamic one and it allows us to generalise this result to the case of arbitrary Renyi entropy.