Electron-phonon coupling in the self-consistent Born approximation of   the t-J model
  We study an undoped t-J model with electron-phonon interaction using the self-consistent Born approximation (SCBA). By neglecting vertex corrections, the SCBA solves a boson-holon model, where a holon couples to phonons and magnons. Comparison with exact diagonalization results for the t-J model suggests that the SCBA describes the electron-phonon interaction fairly accurately over a substantial range of J/t values. Exact diagonalization of the boson-holon model shows that the deviations are mainly due to the neglect of vertex corrections for small J/t and due to the replacement of the t-J model by the boson-holon model for large J/t. For typical values of J/t, the electron-phonon part Sigma_ep of the electron self-energy has comparable contributions from the second order diagram in the electron-phonon interaction and a phonon induced change of magnon diagrams. A very simple approximation to Sigma_ep gives a rather accurate effective mass. Using this approximation, we study the factors influencing the electron-phonon interaction. Typically, we find that the magnons nominally have a stronger coupling to the holon than the phonons. The phonons, nevertheless, drive the formation of small polarons (self-localization) due to important differences between the character of the phonon and magnon couplings. 
O. Gunnarsson and O. R\"osch