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We re-examine the effect of long-range Coulomb interactions on the collectiveamplitude and phase modes in the incommensurate charge-density-wave groundstate of quasi-one-dimensional conductors. Using an effective action approachwe show that the longitudinal acoustic phonon protects the gapless lineardispersion of the lowest phase mode in the presence of long-range Coulombinteractions. Moreover, in Gaussian approximation, amplitude fluctuations arenot affected by long-range Coulomb interactions. We also calculate thecollective mode dispersions at finite temperatures and compare our results withthe measured energies of amplitude and phase modes in K$_{0.3}$MoO$_3$. Withthe exception of the lowest phase mode, the temperature dependence of themeasured mode energies can be quantitatively described within a multi-phononFr\"{o}hlich model for generic electron-phonon interactions neglectinglong-range Coulomb interactions.

Collective modes in the charge-density wave state of K$_{0.3}$MoO$_3$: The role of long-range Coulomb interactions revisited