Competition between ferromagnetic and charge-orbital ordered phases in ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$ for $x=\frac{1}{4},$ $\frac{3}{8},$ and $\frac{1}{2}$

Spin, charge, and orbital structures in models for doped manganites are studied by a combination of analytic mean-field and numerical relaxation techniques. At realistic values for the electron-phonon and antiferromagnetic $t_{2\mathrm{g}}$ spin couplings, a competition between a ferromagnetic (FM) phase and a charge-orbital ordered (COO) insulating state is found for $x=1/4,$ 3/8, and 1/2, as experimentally observed in ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$ for $x=0.3\mathrm{}–\mathrm{0.5.}$ The theoretical predictions for the spin-charge-orbital ordering pattern are compared with experiments. The FM-COO energy difference is surprisingly small for the densities studied, and the results compatible with the presence of a robust colossal-magnetoresistive effect in ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$ in a large density interval.