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

The charge-density-wave (CDW) order with macroscopically occupied electrons distorts the underlying lattice and usually causes the softening of the associated phonon mode. However, previous studies demonstrated that the spin-Peierls transition does not always induce an associated phonon softening, but the central-peak scenario applied in the quasi-one-dimensional compound CuGeO$_3$. We generalize the lattice-dynamics studies on the two-dimensional CDW state at van-Hove-singularity (VHS) filling and find that the CDW ordering could develop a central peak at zero frequency while the associated phonon undergoes hardening. The particle-hole scatterings between VHS points give rise to a low-energy increased charge-density susceptibility, and their coupling to the lattice dynamics induces two poles in the Green function for the CDW-associated phonon mode. The zero-frequency pole corresponds to the collective charge-density and phonon coupling mode. The high-frequency one is related to the high-temperature phonon mode that hardens as reaching the CDW transition. Our result may have the potential implication for the recently discovered Kagome metal $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) in which no soft phonon is observed during the CDW transition.