  We study the one-dimensional $S=1/2$ XXZ model on a finite lattice at zero
temperature, varying the exchange anisotropy $\Delta$ and the number of sites
$N$ of the lattice. Special emphasis is given to the model with $\Delta=1/2$
and $N$ odd, whose ground state, the so-called Razumov-Stroganov state, has a
peculiar structure and no finite-size corrections to the energy per site. We
find that such model corresponds to a special point on the $\Delta$-axis which
separates the region where adding spin-pairs increases the energy per site from
that where the longer the chain the lower the energy. Entanglement properties
do not hold surprises for $\Delta=1/2$ and $N$ odd. Finite-size corrections to
the energy per site non trivially vanish also in the ferromagnetic $\Delta\to
-1^+$ isotropic limit, which is consequently addressed; in this case, peculiar
features of some entanglement properties, due to the finite length of the chain
and related with the change in the symmetry of the Hamiltonian, are evidenced
and discussed. In both the above models the absence of finite-size corrections
to the energy per site is related to a peculiar structure of the ground state,
which has permitted us to provide new exact analytic expressions for some
correlation functions.
