Dilution Effects in Two-Dimensional Quantum Orbital Systems

Interacting orbital degrees of freedom in a Mott insulator are essentially directional and frustrated. In this Letter, the effect of dilution in a quantum-orbital system with this kind of interaction is studied by analyzing a minimal orbital model which we call the two-dimensional quantum compass model. We find that the decrease of the ordering temperature due to dilution is stronger than that in spin models, but it is also much weaker than that of the classical model. The difference between the classical and the quantum-orbital systems arises from the enhancement of the effective dimensionality due to quantum fluctuations.

Figure 1

(a) Temperature dependence of the directional order parameter $\overline{D}(T)$ at $x=0$ for various system sizes $L$. The inset shows $\overline{D}(T)$ at low temperatures obtained in the calculation with the large Trotter numbers (●). ○ are the same data with those in the main panel. (b) $\overline{D}(T)$ for various impurity concentrations.

Figure 2

Temperature dependence of the Binder cumulant $Q$ at $x=0$ for various system sizes $L$. Inset shows temperature dependence of $Q$ at $x=0.06$.

Figure 3

Impurity concentration $x$ dependence of the ordering temperature. ● are for $T_{\mathrm{DO}}$ in the quantum OCM, and ■ are for $T_{\mathrm{CL}}$ in the classical OCM. For comparison, the ordering temperature in the Ising model (▴) are also plotted. Bold arrow indicates the percolation threshold $x_c$ in a square lattice. The inset shows the ordering temperature $T_C$ in the quantum $XY$ model with $S=1/2$ (●) and that in the classical one (■) in a three-dimensional cubic lattice. System size is chosen to be $N={32}^3$ at maximum. The continuous imaginary-time algorithm is utilized.

Figure 4

Contour map of local static correlation function $C^{zz}({\mathbf{r}}_i)$ in a $20\times 20$ site cluster. Four vacancies $X_{1(2)}$ and $Z_{1(2)}$ are represented by ●. Temperature is chosen to be $T/T_{\mathrm{DO}}(x=0)=1.27$.