  We describe quantum entanglement inherent to the polaron ground states of
coupled electron-phonon (or, more generally, particle-phonon) systems based on
a model comprising both local (Holstein-type) and nonlocal (Peierls-type)
coupling. We study this model using a variational method supplemented by the
exact numerical diagonalization on a system of finite size. By way of
subsequent numerical diagonalization of the reduced density matrix, we
determine the particle-phonon entanglement as given by the von Neumann and
linear entropies. Our results are strongly indicative of the intimate
relationship between the particle localization/delocalization and the
particle-phonon entanglement. In particular, we find a compelling evidence for
the existence of a nonanalyticity in the entanglement entropies with respect to
the Peierls-coupling strength. The occurrence of such nonanalyticity -- not
accompanied by an actual quantum phase transition -- reinforces analogous
conclusion drawn in several recent studies of entanglement in the realm of
quantum-dissipative systems. In addition, we demonstrate that the entanglement
entropies saturate inside the self-trapped region where the small-polaron
states are nearly maximally mixed.
