Charge dynamics of t-J model and anomalous bond-stretching phonons in cuprates

The density response of a doped Mott-Hubbard insulator is discussed starting from the t-J model in a slave boson 1/N representation. In leading order O(1) the density fluctuation spectra $N(\mathbf{q},\omega )$ are determined by an undamped collective mode at large momentum transfer, in striking disagreement with results obtained by exact diagonalization, which reveal a very broad dispersive peak, reminescent of strong spin-charge coupling. The 1/N corrections introduce the polaron character of the bosonic holes moving in a uniform RVB background. The resulting $N(\mathbf{q},\omega )$ captures all features observed in diagonalization studies, fulfills the appropriate sum rules, and apart from the broadening of the collective mode shows a new low energy feature at the energy $\chi J+\delta t$ related to the polaron motion in the spinon background. It is further shown that the low energy structure, which is particularly pronounced in $(\pi ,0)$ direction, describes the strong renormalization and anomalous damping of the highest bond-stretching phonons in La${}_{2-x}$Sr${}_x$CuO${}_4$.