Beyond the Hubbard-I solution with a one-pole self-energy at half-filling within the moment approach

We have postulated a single pole for the self-energy, Σ(k,ω), looking for the consequences on the one-particle Green function, G(k,ω) in the Hubbard model. We find that G(k,ω) satisfies the first two sum rules or moments of Nolting (Z. Physik 255 (1972) 25) for any values of the two unknown k parameters of Σ(k,ω). In order to find these two parameters we have used the third and four sum rules of Nolting. G(k,ω) turns out to be identical to the one of Nolting (Z. Physik 255 (1972) 25), which is beyond a Hubbard-I solution since satisfies four sum rules. With our proposal we have been able to obtain an expansion in powers of U for the self-energy (here to second order in U). We present numerical results at half-filling for (1) the static spin susceptibility, χ(T) vs. T/t and (2) the band narrowing parameter, B(T) vs. T/t. The two-pole ansatz of Nolting for the one-particle Green function is equivalent to a single pole ansatz for the self-energy which remains the fundamental quantity for more elaborated calculations when, for example, lifetime effects are included.