Figure 1

(a) Schematic representation of the buckling distortion in the one-dimensional zigzag chain and of the dimerization effect induced by the local interaction (manganeses with opposite spins belong to different chains). (b) Energy gain of the dimerized (ZP) state with respect to the nondimerized (CE) state as a function of $U$ (at $E_{\mathit{JT}}=0$) for an initial distortion $u_0=0.3$. The inset shows equilibrium distortion $u_{\mathit{eq}}$ for the nondimerized state compared to the staggered one found in the ZP state. (c) Difference of the on-site orbital occupancy between the most and least occupied orbitals as a function of $U$ in the dimerized state, on corner (circles) and bridge (triangles) sites. The inset shows the charge disproportion between corner and bridge Mn sites. Dotted lines represent the same quantities in the nondimerized state. (d) Schematic representation of the orbital-order pattern in the dimerized state.Reuse & Permissions

Figure 2

(left) Evolution of total energy as a function of $u_{-}=(u_1-u_2)/2$ at fixed optimum $u_{+}=(u_1+u_2)/2$ at $U=0$ and $U=1$ ($E_{\mathit{JT}}=0$); the inset shows the onset of the two contributions to polarization at $U=1$. (right) Polarization evaluated at $u_0=0.3$ and $E_{\mathit{JT}}=0$ as a function of $U$ (the electron charge $e$ is set to unity).Reuse & Permissions

Figure 3

(left) Difference in on-site orbital occupancy as a function of $E_{\mathit{JT}}$ in the dimerized state on corner (circles) and bridge (triangles) sites. The inset shows the charge disproportion between corner and bridge sites. Dotted lines label the nondimerized results. Open symbols show the same quantities for the breathing mode switched off. (right) Calculated polarization as a function of $E_{\mathit{JT}}$ at $u_0=0.3,U=0$ (solid and open symbols: with and without the breathing mode). The inset shows the energy gain of the dimerized phase with (solid line) and without (dashed line) the breathing mode.Reuse & Permissions

Figure 4

Phase diagram in the $E_{\mathit{JT}}$-$U$ space at (left) $u_0=0.3$ and (right) $u_0=0.25$; ZP is the dimerized state, and CB is the nondimerized one.Reuse & Permissions