  We consider the collective, long-wavelength charge excitations in correlated
media in presence of short- and long-range forces. As an example for the case
of a short-range interaction, we examine the two-dimensional Hubbard model
within dynamical mean-field theory (DMFT). It is shown explicitly that the DMFT
susceptibility including vertex corrections respects the Ward identity and
yields a manifestly gauge-invariant response in finite dimensions. For
computing the susceptibility, we use a different expression and establish its
formal equivalence to the standard DMFT formula. It allows for a more stable
analytical continuation. We find a zero-sound mode expected for short-range
forces. The relation between the vertex corrections, gauge invariance, and the
appearance of the collective modes is discussed. Long-range forces are treated
within extended dynamical mean-field theory. In order to obtain a
gauge-invariant response, it is necessary to additionally incorporate some non
local vertex corrections into the polarization. In doing so, we obtain plasmons
in the three-dimensional Hubbard model. The plasma frequency is determined by
the (single-particle) density distribution as a consequence of gauge
invariance. We compare this result with the plasma frequency extracted from the
analytical continuation of the susceptibility. It is in good agreement with the
prediction from the gauge-invariance condition.
