Spin Excitations in Doped Manganites

We generalize the double exchange to describe holes doped in degenerate e$\text{}_{g}$ orbitals in La$\text{}_{1-x}$A$\text{}_{x}$MnO$\text{}_{3}$ manganites, where A are divalent alkaline earth ions. Assuming an orbital liquid of disordered e$\text{}_{g}$ orbitals we find isotropic ferromagnetic exchange interactions which increase with hole doping. The magnon dispersion agrees very well with the experimental data in ferromagnetic metallic manganites