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

Abstract

Abstract 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 $$t_z$$ t z in the z -direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases $$t_z=0$$ t z = 0 and $$t_z=t$$ t z = t , respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that $$t=1$$ t = 1 , we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, U , and on the hopping parameter $$t_z$$ 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 $$t_z \rightarrow 0$$ t z → 0 . It is found that the asymptotic formulas are qualitatively different for $$t_z = 0$$ t z = 0 (the two-dimensional case) and $$t_z > 0$$ 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 $$t_z \rightarrow 0$$ t z → 0 in which the three-dimensional model reduces to the two-dimensional model.