Impurity scattering in 1D ring of interacting electrons

The stability of the 1D Luttinger liquid phase of interacting spinless particles ($t - V$ model) to an on-site diagonal disorder is studied by exact diagonalization of small 1D rings up to 22 sites. We discuss our numerical results in the light of the phase diagram predicted by the Renormalization Group analysis. In a finite range of attractive interactions (around $V/t \sim -0.5$) the metallic phase (with superconducting fluctuations) seems to be stable up to some small value of the disorder parameter. On the other hand, in the repulsive case, the impurity potential clearly leads to a strong localization of the particles