  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. 
When finite-size corrections vanish: The S=1/2 XXZ model and the   Razumov-Stroganov state