Figure 1

Zero temperature phase diagram for $x=1$ for the two-dimensional DE two orbital model with cooperative Jahn-Teller phonons. The symbols OO and OD stand for orbital ordered and orbital disordered, respectively. $A$, $E$, and $G$ name the different magnetic order defined in the text.Reuse & Permissions

Figure 2

(Color online) Orbital and spin order of $x=1$ manganites in the $x\text{−}y$ plane. Elongated orbitals along the $x$ $\left(y\right)$ directions represent $d_{3x^2-r^2}$ $\left(d_{3y^2-r^2}\right)$ orbitals. Circles represent the Mn ions in a orbital disordered phase. (a) Orbital order present in the ferromagmetic $A$-OO phase. (b) Same as (a) but for the $E$-OO phase. (c) Spin order in the $E$-OD phase. The solid and the dashed lines joining the Mn ions indicate the modulation of the electronic coherence along the zigzag chains. (d) Same as (a) but for the $G$-OO phase. In all the phases there is not modulation of the electric charge and there is an electron located at each Mn ion. The vectors in the different schemes, represent the spatial periodicity in the different phases, $(a,a)$ in the $A$-OO and $G$-OO phases and $(2a,2a)$ in the $E$-OO and $E$-OD phases. In all the figures open and closed symbols represent up and down spins.Reuse & Permissions

Figure 3

Phase diagrams $T\text{−}{J}_{\mathrm{AF}}$ for the two-dimensional DE two orbital model with cooperative Jahn-Teller phonons and $x=1$. In (a) we plot the $\lambda =0$ case and in (b) the $\lambda =1.2t$ case. Continuous lines represent first order transitions whereas dashed lines indicate second order transitions. The abbreviations naming the different phases are explained in the text.Reuse & Permissions

Figure 4

(Color online) Spin order of $x=1$ manganites in the $x\text{−}y$ plane. (a) corresponds to the $E$ phase whereas (c) represents the $A$ phase. The direction perpendicular to the zigzag chains is shown in (a). The averaged magnetization along the $z$ direction for the $E$ and $A$ phases are plotted in (c) and (d), respectively.Reuse & Permissions

Figure 5

Phase diagram $T\text{−}{J}_{\mathrm{AF}}$ as obtained by minimizing the free energy Eq. (18). The parameters of entering in the free energy are obtained by minimizing the microscopic Hamiltonian Eq. (2) for $\lambda =0$ and $x=1$. Continuous lines represent first order transitions whereas dashed lines indicate second order transitions. The shadow region indicates the region where the incommensurate phase exits.Reuse & Permissions

Figure 6

(Color online). Temperature dependence of the magnetic wave vector, $q$, and of the order parameter amplitude, $m_0$ (inset), for two values of $J_{\mathrm{AF}}$. The wave vector $q=\pi ∕2$ corresponds to the $E$ phase and $q=0$ to the $A$ phase. The results are obtained by minimizing the free energy [Eq. (18)].Reuse & Permissions