  We investigate the charge excitations of a Weyl semimetal in the axionic
charge density wave (axionic CDW) state. While it has been shown that the
topological response (anomalous Hall conductivity) is protected against the CDW
state, we find that the long wavelength plasmon excitation is radically
influenced by the dynamics of the CDW order parameter. In the normal state, we
show that an undamped collective mode should exist at q~Q_(CDW) if there is an
attractive interaction favoring the formation of the CDW state. The undamped
nature of this collective mode is attributed to a gap-like feature in the
particle-hole continuum at q~Q_(CDW) due to the chirality of the Weyl nodes,
which is not seen in other materials with CDW instability. In the CDW state,
the long wavelength plasmon excitations become more dispersive due to the
additional interband scattering not allowed in the normal state. Moreover,
because the translational symmetry is spontaneously broken, Umklapp scattering,
the process conserving the total momentum only up to nQ_(CDW) with n an integer
and Q_(CDW) the ordering wave vector, emerges in the CDW state. We find that
the plasmon excitation couples to the phonon mode of the CDW order via the
Umklapp scattering, leading to two branches of resonant collective modes
observable in the density-density correlation function at q~0 and q~Q_(CDW).
Based on our analysis, we propose that measuring these resonant plasmon-axion
excitations around q~0 and q~Q_(CDW) by the momentum-resolved electron energy
loss spectroscopy (M-EELS) could serve as a reliable way to detect the axionic
CDW state in Weyl semimetals.
