  We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on
a finite-size lattice with N sites.We compute the large $N$ behavior of the
spin stiffness, obtaining the correlation length \xi. We use our results to
discuss the scaling behavior of metal-insulator transitions in 1D systems,
taking into account the mapping between the XXZ Heisenberg chain and the
spinless fermion model and known results for the Hubbard model. We study the
scaling properties of both the Hubbard model and the XXZ Heisenberg chain, by
solving numerically the Bethe Ansatz equations. We find that for some range of
values of \xi/N the scaling behavior may be observed for the Hubbard model but
not for the XXZ Heisenserg chain. We show how \xi can be obtained from the
scaling properties of the spin stiffness for small system sizes. This method
can be applied to models having not an exact solution, illuminating their
transport properties.
