Interplay between spin and phonon fluctuations in the double-exchange model for the manganites

We present exact solutions, mainly analytical, for the two-site double-exchange-Holstein model, that allow us to draw a complete picture of the role of both phonon and spin quantum fluctuations in determining the short-range correlations in the manganites. We provide analytical solutions of the model for arbitrary electron-phonon coupling and phonon frequency, for S=1/2 and for the classical spin limit S=infinity, and compare these results with numerical diagonalization of the realistic S=3/2 case. The comparison reveals that the realistic case S=3/2 is not well described by the classical spin limit, which is often used in literature. On the other hand, the phonon fluctuations, parametrized by the phonon frequency omega(0), stabilize ferromagnetic phases with respect to the adiabatic limit. We also provide a complete analysis on the polaron crossover in this model.