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

TL;DR

This paper introduces a single-pole self-energy approach in the Hubbard model that satisfies multiple sum rules, extends previous solutions, and provides numerical results for static spin susceptibility and band narrowing at half-filling.

Contribution

It proposes a novel single-pole self-energy method that satisfies multiple sum rules and extends beyond Hubbard-I, enabling more accurate modeling of correlated electron systems.

Findings

Self-energy satisfies four sum rules, improving upon Hubbard-I.

Derived an U-expansion for the self-energy up to second order.

Numerical results for spin susceptibility and band narrowing at half-filling.

Abstract

We have postulated a single pole for the self-energy, $\Sigma(\vec{k},\omega)$, looking for the consequences on the one-particle Green function, $G(\vec{k},\omega)$ in the Hubbard model. We find that $G(\vec{k},\omega)$ 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 $\Sigma(\vec{k},\omega)$. In order to find these two parameters we have used the third and four sum rules of Nolting. $G(\vec{k},\omega)$ 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, $\chi(T)$ 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.