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Jahn-Teller distortions and the magnetic order in the perovskite manganites

We introduce an effective model for $e_g$ electrons to describe three-dimensional perovskite (La$_{1-x}$Sr$_{x}$MnO$_3$ and La$_{1-x}$Ca$_{x}$MnO$_3$) manganites and study the magnetic and orbital order on $4\times 4\times 4$ cluster using correlated wave functions. The model includes the kinetic energy, and on-site Coulomb interactions for $e_g$ electrons, antiferromagnetic superexchange interaction between $S=3/2$ core spins, and the coupling between $e_g$ electrons and Jahn-Teller modes. The model reproduces the experimentally observed magnetic order: ($i$) $A$-type antiferromagnetic phase in the undoped insulator LaMnO$_3$, with alternating $e_g$ orbitals and with small Jahn-Teller distortions, changing to a conducting phase at 32 GPa pressure, and ($ii$) ferromagnetic order in one-eight doped La$_{7/8}$Sr$_{1/8}$MnO$_3$ and in quarter doped La$_{3/4}$Sr$_{1/4}$MnO$_3$ compounds. For half-doped La$_{1/2}$Ca$_{1/2}$MnO$_3$ one finds a competition between a ferromagnetic conductor and the CE insulating phase; the latter is stabilized by the Jahn-Teller coupling being twice larger than for the strontium-doped compound. Altogether, there is a subtle balance between all Hamiltonian parameters and the phase diagram is quite sensitive to the precise values they take.