Randomly Diluted $e_g$ Orbital-Ordered Systems

Dilution effects on the long-range ordered state of the doubly degenerate $e_g$ orbital are investigated. Quenched impurities without the orbital degree of freedom are introduced in the orbital model where the long-range order is realized by the order-from-disorder mechanism. It is shown by Monte Carlo simulations and the cluster-expansion method that a decrease in the orbital-ordering temperature by dilution is substantially larger than that in the randomly diluted spin models. Tilting of orbital pseudospins around impurities is the essence of this dilution effect. The present theory provides a new viewpoint for the recent resonant x-ray scattering experiments in ${\mathrm{KCu}}_{1-x}{\mathrm{Zn}}_x{\mathrm{F}}_3$.

Figure 1

Impurity-concentration dependence of the normalized orbital-ordering temperature $T_{\mathrm{OO}}(x)/T_{\mathrm{OO}}(x=0)$. The calculated results by the MC method are shown by the solid and open red squares (see text), and those by the CE method for the one- and two-site clusters are shown by the solid and dashed lines, respectively. The results by the CE method where the PS operator is treated as a classical vector are shown by the one-point chain line. For comparison, $T_{\mathrm{N}}$ in the $XY$ (down-pointing triangles) and Ising (stars) models by the MC method, and that in the Heisenberg model (dotted line) by the CE method, are also shown. The inset shows the Zn concentration dependence of $T_{\mathrm{OO}}$ in ${\mathrm{KCu}}_{1-x}{\mathrm{Zn}}_x{\mathrm{F}}_3$ obtained by the RXS experiments [4].

Figure 2

(a) A MC data distribution for the orbital angle $\theta$ in the staggered-type orbital order at $x=0$. The inset shows a data distribution of the order parameter $M_{\mathrm{O}}$ at $x=0$. (b) The finite-size scaling for the correlation length $\xi /L$ as functions of $L^{1/\nu }(T-T_{\mathrm{OO}})/J$ at $x=0$. The inset shows the scaling plot at $x=0.15$.

Figure 3

(a) Temperature dependence of the normalized-order parameter $m_{\mathrm{O}}(x,T)$ [$=M_{\mathrm{O}}/(1-x)$] for various $x$ and $L$. (b) A snapshot in the MC simulation for the PS configuration at $x=0.1$. Open circles indicate the impurities. (c) A snapshot at $x=0.3$.

Figure 4

Impurity-concentration dependence of the normalized orbital-ordering temperature $T_{\mathrm{OO}}(x)/T_{\mathrm{OO}}(x=0)$ with the higher-order JT coupling. The squares, circles, and triangles indicate $T_{\mathrm{OO}}(x)/T_{\mathrm{OO}}(x=0)$ at $J^{\prime }/J=0$, 0.3, and 3, respectively. $T_{\mathrm{OO}}$’s indicated by the open and solid symbols are defined in the text. For comparison, $T_{\mathrm{N}}$ of the $XY$ model (down-pointing triangles) and $T_{\mathrm{OO}}$ at $J^{\prime }/J=0$ by the CE method (solid line) are also plotted.