Polaronic behavior and electron-phonon interaction in cuprates

Photoemission and neutron scattering indicate a substantial electron-phonon coupling in high-$T_c$ cuprates. To address the associated anomalous softening of a half-breathing Cu-O bond-stretching phonon, we derive a $t$-$J$ model with electron-phonon coupling.$^1$ Using input parameters from band structure calculations and solving the model by exact diagonalization, we obtain a good description of the phonon softening.$^1$ We study the interplay of the electron-phonon and Coulomb interactions for a (weakly) doped Mott-Hubbard insulator. Using sum-rules, we find that that the effect of the electron-phonon interaction on the phonon self-energy is strongly suppressed, while there is no corresponding suppression for the electron self-energy or the phonon-induced carrier-carrier interaction.$^2$ Photoemission suggests polaronic behavior in undoped cuprates. Calculating the electron-phonon interaction in a shell model of an undoped cuprate, we find sufficiently strong coupling to give polaronic behavior. Using an adiabatic approximation, we discuss the dispersion and width of the corresponding phonon side-band. {}$^1$O. R\"osch and O. Gunnarsson, Phys. Rev. Lett. {\bf 92}, 146403 (2004); {}$^2$O. R\"osch and O. Gunnarsson, Phys. Rev. Lett. (in press), cond-mat/0407064.