Quantum Behavior of Orbitals in Ferromagnetic Titanates: Novel Orderings and Excitations

We investigate the collective behavior of orbital angular momentum in the spin ferromagnetic state of a Mott insulator with $t_{2g}$ orbital degeneracy. The frustrated nature of the interactions leads to an infinite degeneracy of classical states. Quantum effects select four distinct orbital orderings. Two of them have a quadrupolar order, while the other states show in addition weak orbital magnetism. Specific predictions are made for neutron scattering experiments which might help to identify the orbital order in ${\mathrm{Y}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ and to detect the elementary orbital excitations.

Figure 1

Arrangement of the local quantization axes in states (a) and (b). Each state has four sublattices denoted by numbers. Arrows show a direction of the quantization axes at each site. They represent also a snapshot of local correlations of orbital angular momentum: on every bond, two out of three components $l_{\alpha }$ are correlated antiferromagnetically.

Figure 2

$t_{2g}$ electron density (left) and intensity of the orbital contribution to the structure factor $S(\overrightarrow{q},\omega )$ (right). Upper panel: Quadrupole ordered state I(a). Because of the more spatial anisotropy of electron distribution, this state may be preferred by electron-lattice coupling. “Hot spot” at $(\pi ,\pi ,\pi )$ point is due to coherency factors stemming from four sublattice structure. Lower panel: Magnetic state II(a). Shown in the inset are the expected orbital Bragg peak positions.

Figure 3

(a) Energy difference (in units of $J_{\mathrm{S}\mathrm{E}}^{(0)}$) between AF and ferromagnetic states as a function of $\theta$. The solid line is obtained within the SE model. Parameters are $\eta =0.12$ and $U/{\Delta }_{\mathrm{c}\mathrm{r}}=2.5$. The broken line includes small JT energy gain in the orbitally ordered ferromagnetic state. (b) Qualitative picture of the evolution of interactions in the spin channel. An average, static component of the spin exchange “constant” seen by coherent spin waves is represented by a solid curve. Its sign depends on local correlations of orbitals. To the right of the critical point (QCP) these correlations are more antiferromagnetic, supported by noncollinear orbital orderings. To the left, the genuine ground state of the $t_{2g}$ system on cubic lattice, an orbital disordered state supporting spin AF, is stabilized. In the proximity area, a fluctuating part of the overall SE interaction dominates, and separation of the spin and orbital degrees of freedom might no longer be possible.