Absence of Antiferromagnetic Ordering in Hubbard's Simple Decoupling Scheme

The self-consistency criterion for the antiferromagnetic solution of a single-band Hubbard model in Hubbard's simple decoupling scheme is, here, analyzed in detail as a function of the number n (0<n<2) of electrons per site. It is shown that a self-consistent antiferromagnetic solution exhibiting a first- or a second-order phase transition does not exist for any value of n. For n=1 the only possible solution at any temperature is shown to be the one with zero sublattice magnetization. These results are contrary to a claim by Arai.