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

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Abstract
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 greater-than-or-equal-to 3 irrelevant in the weak-coupling limit, while for d = 1 and d = 2 it strongly renormalizes the Hartree-Fock result, although it does not 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-1ln2(m-1)] for d = 2. 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.