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The charge-density-wave (CDW) order with macroscopically occupied electronsdistorts the underlying lattice and usually causes the softening of theassociated phonon mode. However, previous studies demonstrated that thespin-Peierls transition does not always induce an associated phonon softening,but the central-peak scenario applied in the quasi-one-dimensional compoundCuGeO$_3$. We generalize the lattice-dynamics studies on the two-dimensionalCDW state at van-Hove-singularity (VHS) filling and find that the CDW orderingcould develop a central peak at zero frequency while the associated phononundergoes hardening. The particle-hole scatterings between VHS points give riseto a low-energy increased charge-density susceptibility, and their coupling tothe lattice dynamics induces two poles in the Green function for theCDW-associated phonon mode. The zero-frequency pole corresponds to thecollective charge-density and phonon coupling mode. The high-frequency one isrelated to the high-temperature phonon mode that hardens as reaching the CDWtransition. Our result may have the potential implication for the recentlydiscovered Kagome metal $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) in which no softphonon is observed during the CDW transition.

Lattice dynamics in the charge-density-wave metal at a van-Hove-singularity filling