Evolution of Fermi Liquid Behavior with Doping in the Hubbard Model

We calculate the single-particle Green's function for the tight-binding band structure, $\xi_{\vec p}=-2t\cos p_x-2t\cos p_y -\mu$, with a function of chemical potential $\mu$ for square-lattice system. The form of the single-particle self-energy, $\Sigma({\vec p}, E)$, is determined by the density-density correlation function, $\chi({\vec q}, \omega)$, which develops two peaks for $\mu \gtrsim -2.5t$ unlike parabolic band case. Near half filling $\chi({\vec q}, \omega)$ becomes independent of $\omega$, one dimensional behavior, at intermediate values of $\omega$ which leads to one dimensional behavior in $\Sigma({\vec p},E)$. However $\mu \leq -0.1t$ there is no influence on the Fermi Liquid dependences from SDW instability. The strong $\vec p$ and $E$ dependence of the off-shell self-energy, $\Sigma(p,E)$, found earlier for the parabolic band is recovered for $\mu \lesssim -t$ but deviations from this develop for $\mu \gtrsim -0.1t$. The resonance peak width of the spectral function, $A({\vec p}, E)$ has linear dependence in $\xi_{\vec p}$ due to the $E$ dependence of the imaginary part of $\Sigma({\vec p}, E)$. We point out that an accurate detailed form for $\Sigma({\vec p},E)$ would be very difficult to recover from ARPES data for the spectral density.