  Recent spectroscopic measurements in a number of strongly correlated metals
that exhibit non-Fermi liquid like properties have observed evidence of
anomalous frequency and momentum-dependent charge-density fluctuations.
Specifically, in the strange metallic regime of the cuprate superconductors,
there is a featureless particle-hole continuum exhibiting unusual power-laws,
and experiments suggest that the plasmon mode decays into this continuum in a
manner that is distinct from the expectations of conventional Fermi liquid
theory. Inspired by these new experimental developments, we address the nature
of low-energy collective modes and the particle-hole continua for different
"solvable" lattice models of non-Fermi liquids that host a critical Fermi
surface -- a sharp electronic Fermi surface without any low-energy electronic
quasiparticles. We scrutinize theoretically the possible existence of a
long-lived zero-sound mode, which is renormalized to the plasma frequency in
the presence of long-ranged coulomb interactions, and its decay into the
continuum over a wide range of frequencies and momenta. Quite remarkably, some
of the models analyzed here can account for certain aspects of the universal
experimental observations, that clearly lie beyond the purview of standard
Fermi liquid theory.
