Numerical study of $t_{2g}$ orbital system with ferromagnetic polarization

Finite-temperature orbital state in a ferromagnetic Mott insulator with triply degenerate $t_{2g}$ orbital is investigated numerically. We employ the quantum Monte Carlo simulation with the loop algorithm. Indications for conventional staggered-type orbital order are not remarkable down to the lowest temperature to which the present simulation can get access. Physical parameters monitoring the off-diagonal orbital order, which is characterized by a linear combination of the $\left(d_{yz},d_{zx},d_{xy}\right)$ orbital-wave functions with equal weights, are not conspicuous. It is found that an orbital gaplike behavior appears in the uniform orbital susceptibility. This is supported by a threshold behavior in the staggered correlation function in a calculation with the additional Ising-type interaction. Some rigorous remarks for the long-range orbital order are also presented.

Figure 1

(Color online) (a) Staggered correlation function $M^x$ for the orbital PS operator $T_{iz}^x$. (b) Staggered orbital susceptibility ${\chi }^x$. For comparison, $\sqrt{{\chi }^{x}T/L^3}$ is also plotted in (a). The inset of (b) shows specific heat $C$.

Figure 2

(Color online) (a) Correlation function $G^Q$ for the off-diagonal orbital operator $Q_i$. (b) Correlation function $G^{l\left(m\right)}$ for the orbital PS operator $T_{iz}^x$. The system size in (b) is taken to be $L=10$. Results in $G^{x\left(3\right)}$ and $G^{x\left(4\right)}$ are multiplied by 5. The inset of (a) shows the dimer order parameter $D^x$.

Figure 3

(Color online) (a) Uniform susceptibility for the orbital PS operator $T_{iz}^x$. (b) Correlation function $M^z$ calculated in the Hamiltonian $\mathcal{H}+{\mathcal{H}}^{\prime }$. The system size and the Trotter number in (b) are chosen to be $L=8$ and $n=10$, respectively. The inset of (b) shows the correlation function $K^x$ for NN PS operators.