  The low-energy physics of the one-dimensional Pair-Hopping (PH) and
attractive Hubbard models are expected to be similar. Based on numerical
calculations on small chains, several authors have recently challenged this
idea and predicted the existence of a phase transition at half-filling and
finite positive coupling for the pair-hopping model. We re-examine the
controversy by making systematic comparisons between numerical results obtained
for the PH and attractive Hubbard models. To do so, we have calculated the
Luttinger parameters (spin and charge velocities, stiffnesses, etc...) of the
two models using both the Density Matrix Renormalization Group method for large
systems and Lancz\'os calculations with twisted boundary conditions for smaller
systems. Although most of our results confirm that both models are very similar
we have found some important differences in the spin properties for the small
sizes considered by previous numerical studies (6-12 sites). However, we show
that these differences disappear at larger sizes (14-42 sites) when
sufficiently accurate eigenstates are considered. Accordingly, our results
strongly suggest that the ground-state phase transition previously found for
small systems is a finite size artefact. Interpreting our results within the
framework of the Luttinger liquid theory, we discuss the origin of the apparent
contradiction between the predictions of the perturbative Renormalization group
approach and numerical calculations at small sizes.
