Towards a generalized hydrodynamics description of Renyi entropies in integrable systems

We investigate the steady-state Renyi entanglement entropies after a quench from a piecewise homogeneous initial state in integrable models. In the quench protocol, two macroscopically different chains (leads) are joined together at the initial time, and the subsequent dynamics is studied. We study the entropies of a finite subsystem at the interface between the two leads. The density of Renyi entropies coincides with that of the entropies of the generalized Gibbs ensemble that describes the interface between the chains. By combining the generalized hydrodynamics treatment of the quench with the Bethe ansatz approach for the Renyi entropies, we provide exact results for quenches from several initial states in the anisotropic Heisenberg chain (XXZ chain), although the approach is applicable, in principle, to any low-entangled initial state and any integrable model. An interesting protocol that we consider is the expansion quench, in which one of the two leads is prepared in the vacuum of the model excitations. An intriguing feature is that for moderately large anisotropy the transport of bound state is not allowed. Moreover, we show that there is a "critical" anisotropy, below which bound-state transport is permitted. This is reflected in the steady-state entropies, which for large enough anisotropy do not contain information about the bound states. Finally, we benchmark our results against time-dependent density matrix renormalization group simulations.