Study of Pairing Correlations in the Attractive Hubbard Model on Chains, Ladders, and Squares

We report the results of zero temperature quantum Monte Carlo simulations and zero temperature mean-field calculations of the attractive Hubbard model on chains, ladders, and square lattices. We investigated the predictability of the BCS approximation, the dimensional cross-over of the pairing correlation function from one to two dimensions as a function of the ladder width, and the scaling of these correlations to the thermodynamic limit of the two-dimensional model. We found that the BCS wave function is quantitatively correct only for small values of $U/t$. For the system sizes, electron fillings, and interaction strengths studied, we never saw the dimensional cross-over. In general our ability to achieve the dimensional cross-over and accurate scaling to the thermodynamic limit was limited by the size of the systems we could simulate. For these sizes, although we saw the necessary signature of ODLRO, the properties of the model did not vary monotonically with increasing system size because of shell effects. We contrast this situation with the dimensional cross-over and scaling to the thermodynamic limit of the Ising model.