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 $S=\infty$, 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.