Spin-Orbital Entanglement and Violation of the Goodenough-Kanamori Rules

We point out that large composite spin-orbital fluctuations in Mott insulators with $t_{2g}$ orbital degeneracy are a manifestation of quantum entanglement of spin and orbital variables. This results in a dynamical nature of the spin superexchange interactions, which fluctuate over positive and negative values, and leads to an apparent violation of the Goodenough-Kanamori rules.

Figure 1

Intersite spin $S_{ij}$ (filled circles), orbital $T_{ij}^{(t,e)}$ (empty circles), and composite spin-orbital $C_{ij}^{(t,e)}$ (crosses) correlations as functions of Hund’s exchange $\eta S$, for $V=0$ (left) and for $V=J$ (right) for: (a), (d) $d^1$ model, (b), (e) $d^2$ model, and (c), (f) $d^9$ model.

Figure 2

Spin exchange constants $J_{ij}$ at $V=0$ (left) and at $V=J$ (right) as functions of Hund’s exchange $\eta S$ for (a), (d) $d^1$ model; (b), (e) $d^2$ model; (c), (f) $d^9$ model. In the shaded regions in (a), (b), (d), and (e), $J_{ij}$ is negative (FM) and yet the spin correlations are AF, $S_{ij}<0$ (see Fig. 1).