Effect of Electron Correlation on Phonon Spectra in Cuprates

We examine theoretically the phonon dynamics in the electron-phonon-coupled systems. The model Hamiltonian is a one-dimensional Hubbard model where the hopping of electrons induces the electron-phonon coupling. The numerically exact diagonalization of the Hamiltonian with truncation of the total number of phonons is performed. We calculate the phonon excitation spectra and find that the softening of phonons occurs in insulating case, and a diffusive character appears in the metallic case. We also calculate the charge and spin excitation spectra and find that the dynamics of the phonon coupled with the hopping of electrons is affected by the low energy charge and spin excitations.