Ferromagnetic, $\mathit{A}$-Type, and Charge-Ordered $\mathrm{CE}$-Type States in Doped Manganites Using Jahn-Teller Phonons

The two-orbital model for manganites with both noncooperative and cooperative Jahn-Teller phonons is studied at hole density $x=0.5\mathrm{}$ using Monte Carlo techniques. The phase diagram is obtained by varying the electron-phonon coupling and the $t_{2g}$-spins exchange. The insulating $\mathrm{CE}$-type charge- and orbital-ordered state with the $z$-axis charge stacking observed in narrow-bandwidth manganites is stabilized in the simulations. Its charge gap ${\Delta }_{\mathrm{CO}}$ is much larger than the critical temperature $k_{\mathrm{B}}{T}_{\mathrm{CO}}$. Metalliclike $A$-type and ferromagnetic states are also obtained in the same framework, and the phase boundaries among them have first-order characteristics.