Phonon softening and dispersion in the 1D Holstein model of spinless   fermions
  We investigate the effect of electron-phonon interaction on the phononic properties in the one-dimensional half-filled Holstein model of spinless fermions. By means of determinantal Quantum Monte Carlo simulation we show that the behavior of the phonon dynamics gives a clear signal of the transition to a charge-ordered phase, and the phase diagram obtained in this way is in excellent agreement with previous DMRG results. By analyzing the phonon propagator we extract the renormalized phonon frequency, and study how it first softens as the transition is approached and then subsequently hardens in the charge-ordered phase. We then show how anharmonic features develop in the phonon propagator, and how the interaction induces a sizable dispersion of the dressed phonon in the non-adiabatic regime. 
C.E. Creffield, G. Sangiovanni, and M. Capone