We investigate the steady-state R\'enyi 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 R\'enyi entropies coincides with that of the entropies of the
Generalized Gibbs Ensemble (GGE) that describes the interface between the
chains. By combining the Generalized Hydrodynamics (GHD) treatment of the
quench with the Bethe ansatz approach for the R\'enyi 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, above
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 (tDMRG) simulations.Comment: 18 pages, 8 figures, similar to published versio
