Effects of quasiparticle screening on collective modes: Incommensurate charge-density-wave systems

By including the Coulomb interaction term in the usual Fr\"olich Hamiltonian, the dispersion of the phase mode in incommensurate charge-density-wave systems is found. In the long-wavelength limit, the spectrum consists of an optic branch and an acoustic branch, with the optic branch increasingly dominant at low temperatures and the acoustic branch dominant at high temperatures. The optic-mode frequency is enhanced because of the small effective mass of the quasiparticles. For increasing wave numbers, the two branches merge together, beyond which the single branch approaches an optic frequency ((3/2)1/2{{ω}}_{{α}}$, where ${{ω}}_{{α}}$ is the amplitude-mode frequency. In the acoustic branch, the charge densities of the condensed electrons and the quasiparticles oscillate out of phase, balancing each other, while in the optic branch, both components oscillate in phase.