Spin gap and superconductivity in the three-dimensional attractive   Hubbard model
  We study the phase diagram for the attractive ({\it i.e.,} negative-$U$) Hubbard model on a simple cubic lattice, through Monte Carlo simulations. We obtain the critical temperature, $T_c$, for superconductivity from a finite-size scaling analysis of the data for the pairing correlations. For fixed on-site attraction, $U$, $T_c$ displays a maximum near the filling factor 0.9, roughly independent of $U$. For fixed filling we estimate the crossover temperature $T^\times(U)$, separating the normal states: metallic and spin-gap. There is also a critical value $U_p$ for pair formation, the magnitude of which seems to be independent of doping. The relevance of these results to the high-$T_c$ oxides is discussed. 
Raimundo R dos Santos