One-dimensional pair-hopping and attractive Hubbard models: A   comparative study
  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. 
Mathias van den Bossche and Michel Caffarel