Metal-Insulator Transition in the Two-Dimensional Hubbard Model at Half-Filling with Lifetime Effects within the Moment Approach

We explore the effect of the imaginary part of the self-energy, $Im\Sigma(\vec{k},\omega)$, having a single pole, $\Omega(\vec{k},\omega)$, with spectral weight, $\alpha(\vec{k})$, and quasi-particle lifetime, $\Gamma(\vec{k})$, on the density of states. We solve the set of parameters, $\Omega(\vec{k},\omega$), $\alpha(\vec{k})$, and $\Gamma(\vec{k})$ by means of the moment approach (exact sum rules) of Nolting. Our choice for $\Sigma(k,\omega)$, satisfies the Kramers - Kronig relationship automatically. Due to our choice of the self - energy, the system is not a Fermi liquid for any value of the interaction, a result which is also true in the moment approach of Nolting without lifetime effects. By increasing the value of the local interaction, $U/W$, at half-filling ($\rho = 1/2$), we go from a paramagnetic metal to a paramagnetic insulator, (Mott metal - insulator transition ($MMIT$)) for values of $U/W$ of the order of $U/W \geq 1$ ($W$ is the band width) which is in agreement with numerical results for finite lattices and for infinity dimensions ($D = \infty$). These results settle down the main weakness of the spherical approximation of Nolting: a finite gap for any finite value of the interaction, i.e., an insulator for any finite value of $U/W$. Lifetime effects are absolutely indispensable. Our scheme works better than the one of improving the narrowing band factor, $B(\vec{k})$, beyond the spherical approximation of Nolting.