Dominant particle-hole contributions to the phonon dynamics in the spinless one-dimensional Holstein model

In the spinless Holstein model at half-filling the coupling of electrons to phonons is responsible for a phase transition from a metallic state at small coupling to a Peierls distorted insulated state when the electron-phonon coupling exceeds a critical value. For the adiabatic case of small phonon frequencies, the transition is accompanied by a phonon softening at the Brillouin zone boundary whereas a hardening of the phonon mode occurs in the anti-adiabatic case. The phonon dynamics studied in this letter do not only reveal the expected renormalization of the phonon modes but also show remarkable additional contributions due to electronic particle-hole excitations.