Collective modes in the paramagnetic phase of the Hubbard model

The charge dynamical response function of the Hubbard model is investigated on the square lattice in the thermodynamical limit. The obtained charge excitation spectra consist of a continuum, a gapless collective mode with anisotropic zero-sound velocity, and a correlation induced high-frequency mode at $\omega\approx U$. The correlation function is calculated from Gaussian fluctuations around the paramagnetic saddle-point within the Kotliar and Ruckenstein slave-boson representation. Its dependence on the on-site Coulomb repulsion $U$ and density is studied in detail. An approximate analytical expression of the high frequency mode, that holds for any lattice with one atom in the unit cell, is derived. Comparison with numerical simulations, perturbation theory and the polarization potential theory is carried out. We also show that magnetic instabilities tend to vanish for $T\gtrsim t/6$, and finite temperature phase diagrams are established.