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${}_{20}$

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 ${\mathrm{\Gamma }}_3$ 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.