  The realization of antiferromagnetic (AF) correlations in ultracold fermionic
atoms on an optical lattice is a significant achievement. Experiments have been
carried out in one, two, and three dimensions, and have also studied
anisotropic configurations with stronger tunneling in some lattice directions.
Such anisotropy is relevant to the physics of cuprate superconductors and other
strongly correlated materials. Moreover, this anisotropy might be harnessed to
enhance AF order. Here we numerically investigate, using Determinant Quantum
Monte Carlo, a simple realization of anisotropy in the 3D Hubbard model in
which the tunneling between planes, $t_\perp$, is unequal to the intraplane
tunneling $t$. This model interpolates between the three-dimensional isotropic
($t_\perp = t$) and two-dimensional ($t_\perp =0$) systems. We show that at
fixed interaction strength to tunneling ratio ($U/t$), anisotropy can enhance
the magnetic structure factor relative to both 2D and 3D results. However, this
enhancement occurs at interaction strengths below those for which the N\'eel
temperature $T_{\rm N\acute{e}el}$ is largest, in such a way that the structure
factor cannot be made to exceed its value in isotropic 3D systems at the
optimal $U/t$. We characterize the 2D-3D crossover in terms of the magnetic
structure factor, real space spin correlations, number of doubly-occupied
sites, and thermodynamic observables. An interesting implication of our results
stems from the entropy's dependence on anisotropy. As the system evolves from
3D to 2D, the entropy at a fixed temperature increases. Correspondingly, at
fixed entropy, the temperature will decrease going from 3D to 2D. This suggests
a cooling protocol in which the dimensionality is adiabatically changed from 3D
to 2D.
