Dynamical susceptibility of a Fermi liquid

We study the dynamic response of a Fermi liquid in the spin, charge and nematic channels beyond the random phase approximation for the dynamically screened Coulomb potential. In all the channels, one-loop order corrections to the irreducible susceptibility result in a non-zero spectral weight of the corresponding fluctuations above the particle-hole continuum boundary. It is shown that the imaginary part of the spin susceptibility, $\text{Im}\chi_{s}(\bf{q},\omega)$, falls off as $q^2/\omega$ for frequencies above the continuum boundary ($\omega\gg v_{F} q$) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, $\text{Im}\chi_c(\bf{q},\omega)$, falls off as $(q/k_F)^2 q^2/\omega$ for frequencies above the plasma frequency. An extra factor of $(q/k_F)^2$ in $\text{Im}\chi_c(\bf{q},\omega)$ as compared to $\text{Im}\chi_{s}(\bf{q},\omega)$ is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with $\omega$ up to a peak at the ultraviolet energy scale-- the plasma frequency and/or Fermi energy--and then decreases with $\omega$. We also obtain explicit forms of the spin susceptibility from the kinetic equation in the collisionless limit and for the Landau function that contains up to first three harmonics.