Universality of Mean-Field Antiferromagnetic Order in an Anisotropic 3D Hubbard Model at Half-Filling

We study Hartree–Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x- and y-directions and a possibly different hopping parameter \(t_z\) in the z-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases \(t_z=0\) and \(t_z=t\), respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that \(t=1\), we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, U, and on the hopping parameter \(t_z\). We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as \(t_z \rightarrow 0\). It is found that the asymptotic formulas are qualitatively different for \(t_z = 0\) (the two-dimensional case) and \(t_z > 0\) (the case of nonzero hopping in the z-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit \(t_z \rightarrow 0\) in which the three-dimensional model reduces to the two-dimensional model.