Bond stretching phonon softening and isotope effect in a phenomenological model for cuprate superconductors

Lattice and charge degrees of freedom of cuprate superconductors remain an intensively debated topic, both experimentally and theoretically, because of the strong anomalies observed in some of the phonon branches. In particular, the bond-stretching phonon modes show a knee like softening and a large damping for momenta around 0.25-0.3 r.l.u. along, e.g., (1, 0, 0) direction of the Brillouin zone. We discuss a phenomenological model where these anomalies are caused by coupling to electron charge fluctuations and explore a possibility to extract the spectrum of the latter from the phonon data. In particular, we predict a significant deviation from the standard value of the oxygen isotope substitution effect on the phonon spectrum itself. The phonon dispersion and linewidth are shown to contain complementary information related to the isotope effect which could allow a detailed description of the electronic spectrum.