We consider the t-J model for doped manganites, and show that double exchange for correlated e g electrons favors in the ferromagnetic metallic phase an orbital liquid ͑i.e., orbital-disordered͒ state. Therefore the spinwave stiffness is isotropic, and increases with hole doping x, providing a direct measure of the kinetic energy ϰx of strongly correlated e g electrons. The superexchange interactions, which stabilize orbital order at low doping, are frustrated in the orbital liquid and only reduce the stiffness constant, leading to a quantitative explanation of the magnon dispersion. DOI: 10.1103/PhysRevB.65.052414 PACS number͑s͒: 75.30.Vn, 71.27.ϩa, 75.30.Ds, 75.30.Et Strong Coulomb interactions in transition-metal oxides are responsible for numerous fascinating phenomena. They suppress charge fluctuations in the Mott-Hubbard insulators, and replace them by superexchange ͑SE͒ interactions. While these interactions are antiferromagnetic ͑AF͒ when they follow just from the Pauli principle for a single orbital, they have a richer structure when degenerate d orbitals are involved, as in cuprates or manganites. It is evident that in such situations the orbital and spin degrees of freedom have to be considered on equal footing, 1 and the SE interactions are highly frustrated even on a cubic lattice. This was recently recognized as the origin of interesting quantum effects both for e g and t 2g electrons. 2 However, in undoped e g systems such frustration is likely removed by the Jahn-Teller ͑JT͒ effect, which leads to a structural phase transition and stabilizes orbital ordering. The best example is undoped LaMnO 3 with A-type AF spin order which coexists with orbital order, 3,4 and can be understood only as a superposition of the cooperative JT effect and the SE. 5 The theoretical understanding of doped manganese oxides La 1Ϫx A x MnO 3 , with AϭSr,Ca,Pb, is among the most challenging current areas of research in condensed-matter physics. Experimental studies have revealed their rich phase diagrams, 3 originating from the competition between charge, spin, and orbital degrees of freedom. Charge fluctuations are thereby partly suppressed by the large Coulomb interaction UӍ5.9 eV ͑Ref. 6͒ between two e g electrons on the same ion. In addition, the Hund's exchange J H Ӎ0.7 eV couples the sϭ1/2 spins of the e g electrons strongly to the S t ϭ3/2 core spins of the t 2g electrons, resulting in large Sϭ2 spins of Mn 3ϩ ions. When LaMnO 3 is doped, holes that are created in e g orbitals can move but only when all spins are parallel, as explained by the double-exchange ͑DE͒ mechanism, 7 thus leading to a ferromagnetic ͑FM͒ state. When the orbital degeneracy is ignored, as in the original DE theory, isotropic magnon excitations are obtained. 8 However, when the orbital degrees of freedom are taken into account, one finds that the kinetic energy is lowered when orbital order occurs and the system becomes effectively lower dimensional. 9-11 Such states have broken cubic symmetry, and thus anisotropic magnetic excitations. 11...