Onsager-reaction-field correction in the half-filled Hubbard model

The Onsager-reaction-field correction to Hartree-Fock theory recently derived by Georges and Yedidia [Phys. Rev. B 43, 3475 (1991)] is evaluated analytically for the half-filled positive-U Hubbard model on a d-dimensional hypercubic lattice. It is shown that the Onsager correction is for all d{≥}3 irrelevant in the weak-coupling limit, while for d=1 and d=2 it strongly renormalizes the Hartree-Fock result, although it does note completely destabilize the spin-density wave. We examine the perturbation theory for the energy change induced by a small staggered magnetization m to all orders in U, and show that one obtains an infinite series in powers of (U/t)ln[m^-1ln(m^-1)] for d=1, and in powers of (U/t)ln²[m^-1{ln}²$(${m}^{{{-}}1}$)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.