  Recent experiments with ultracold fermionic atoms in optical lattices have
provided a tuneable and clean realization of the attractive Hubbard model
(AHM). In view of this, several physical properties may be thoroughly studied
across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong
(Bose-Einstein condensation, BEC) couplings. Here we report on extensive
determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice,
from which several different quantities have been calculated and should be
useful as a roadmap to experiments. We have obtained a detailed phase diagram
for the critical superconducting temperature, $T_c$, in terms of the band
filling, $\ave{n}$, and interaction strength, $U$, from which we pinpoint a
somewhat wide region $|U|/t \approx 5 \pm 1$ ($t$ is the hopping amplitude) and
$\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum $T_c \approx 0.16 t$. Two
additional temperature scales, namely pairing, $T_p$, and degeneracy, $T_d$,
have been highlighted: the former sets the scale for pair formation (believed
to be closely related to the scale for the gap of spin excitations in
cuprates), while the latter sets the scale for dominant quantum effects. Our
DQMC data for the distribution of doubly occupied sites, for the momentum
distribution function, and for the quasiparticle weight show distinctive
features on both sides of the BCS-BEC crossover, being also suggestive of an
underlying crossover between Fermi- and non-Fermi liquid behaviors.
