Successive single-q and double-q orders in an anisotropic XY model on the diamond structure: a model for quadrupole ordering in PrIr$_2$Zn$_{20}$

Abstract

Quadrupole ordering with the ordering wavevector at the L points in PrIr$_2$Zn$_{20}$ under magnetic fields is analyzed using classical Monte Carlo simulations based on an effective $Γ_3$ quadrupole model on the diamond structure. We demonstrate that competition between the magnetic field and quadrupole anisotropy leads to a rich phase diagram for magnetic fields applied parallel to [001], which includes switching between a single-q state and a double-q state. We also show that a symmetry-allowed biquadratic intersite interaction, corresponding to a hexadecapole interaction, is crucial for reproducing the weak-field topology observed in experiments.

Figures (9)

• Figure 1: (a) Diamond structure with the sites for the sublattice $A$ ($B$) being indicated in green (orange). Exchange interactions $J_{1,2,3,4}$ and $K$ are indicated. (b) The first Brillouin zone. the L points ${\bm{k}} _{1,2,3,4}$ are indicated by green circles.
• Figure 2: (a) Specific heat $C$ as a function of $T$ for $(J_2,J_4,K,h)=(0,1,0,0)$ and $L=4$--$10$. Inset: single-$\bm{q}$ structure factor $S_{ {\bm{k}} }$. (b) Double-$\bm{q}$ structure factor $D_{ {\bm{k}} }$. (c) Extrapolation of $D_{ {\bm{k}} }$ to $L\to \infty$ for $T=2.5$. See the vertical dashed line in (b). Each straight line is a fit for $L$ and $L+2$ up to $L=12$.
• Figure 3: Order parameter angle distributions for $b=0.5$, $h=0$, $J_2=K=0$, and $J_4=1.0$. The system size is $L=8$ and $2\times 10^4$ MCSs are used. (a) $\theta_\Gamma$ and $\theta_{\rm L}$ and (b) $\theta_\Gamma-\theta_{\rm L}$ for $T=1.0$, where the collinear single-$\bm{q}$ order is realized. (c) $\theta_\Gamma$, $\theta_{\rm L}$, and $\theta_{\rm L'}$ and (d) $\theta_\Gamma-\theta_{\rm L}$, $\theta_\Gamma-\theta_{\rm L'}$, $\theta_{\rm L}-\theta_{\rm L}$ for $T=0.4$, where the double-$\bm{q}$ order is realized. In the inset in (b) and (d), schematic configurations of order parameters are illustrated for typical selected domains.
• Figure 4: $T$ dependence of anisotropic structure factor (a) $S_{ {\bm{k}} x}$ and (b) $S_{ {\bm{k}} y}$ for $h=-1.0$, $b=0.5$, $J_2=K=0$, and $J_4=1.0$ with $L=4$, 6, 8, and 10. (c) Phase diagram for $\bm{H} \parallel [110]$ ($h<0$). The color map is $\Delta_{1 \bm{q} }=S_{ {\bm{k}} x}-S_{ {\bm{k}} y}$ for $L=8$. The transition temperature for $S_y(\ell)$ is evaluated by the Binder ratio and indicated by circles (cyan). Inset in (a): Schematic order parameter configuration in the canted single-$\bm{q}$ phase labeled as 1$\bm{q}$(ca).
• Figure 5: Double-$\bm{q}$ (top), single-$\bm{q}$$S_{ {\bm{k}} x}$ (middle), and $S_{ {\bm{k}} y}$ (bottom) structure factors for (a) $h=1$ and (b) $h=8$, with $L=4,6,8$, and 10. The other parameters are the same as those in Fig. \ref{['fig:4']}. Vertical dashed line represents the phase boundaries as a guid for eyes. Schematic order parameter configurations are also shown as in Fig. \ref{['fig:3']}.