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Beyond the Hubbard-I Solution with a One-Pole Self-Energy at Half-Filling with the Moment Approach: Non-Linear Effects

We have postulated a single pole for the self-energy, $Σ(\vec{k},ω)$, looking for the consequences on the one-particle Green function, $G(\vec{k},ω)$ in the Hubbard model. We find that $G(\vec{k},ω)$ satisfies the first two sum rules or moments of Nolting (Z. Physik 255, 25 (1972)) for any values of the two unknown $\vec{k}$ parameters of $Σ(\vec{k},ω)$. In order to find these two parameters we have used the third and four sum rules of Nolting. $G(\vec{k},ω)$ turns out to be identical to the one of Nolting (Z. Physik 225, 25 (1972)), 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.