  The study of charge-density wave (CDW) distortions in Weyl semimetals has
recently returned to the forefront, inspired by experimental interest in
materials such as (TaSe4)2I. However, the interplay between collective phonon
excitations and charge transport in Weyl-CDW systems has not been
systematically studied. In this paper, we examine the longitudinal
electromagnetic response due to collective modes in a Weyl semimetal gapped by
a quasi one-dimensional charge-density wave order, using both continuum and
lattice regularized models. We systematically compute the contributions of the
collective modes to the linear and nonlinear optical conductivity of our
models, both with and without tilting of the Weyl cones. We discover that,
unlike in a single-band CDW, the gapless CDW collective mode does not
contribute to the conductivity unless the Weyl cones are tilted. Going further,
we show that the lowest nontrivial collective mode contribution to charge
transport with untilted Weyl cones comes in the third-order conductivity, and
is mediated by the gapped amplitude mode. We show that this leads to a sharply
peaked third harmonic response at frequencies below the single-particle energy
gap. We discuss the implications of our findings for transport experiments in
Weyl-CDW systems.
