Charge density wave and phonon softening in the Holstein model: A Quantum Monte Carlo study

We present a Quantum Monte Carlo study of electron-phonon dynamics in the two dimensional Holstein model on a square lattice. We concentrate here on the softening of the phonon modes as the coupling between the electrons and the phonons is varied. Information about the instability of the electronic system toward the formation of a charge-density wave is also extracted from the phonon correlation functions. We also extract information about the phonon spectral densities using the Maximum Entropy method. Results are presented for various values of the wave-vectors in the two dimensional lattice. An intriguing softening of the phonon frequency at vanishing wave-vector is presented. We also present the effect of anharmonicity on the dynamics of the Holstein model.