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 $Γ_3$ 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.

💡 Research Summary

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This paper investigates the quadrupolar ordering observed in the cubic 1‑2‑20 compound PrIr₂Zn₂₀ by means of classical Monte‑Carlo simulations of an effective Γ₃ quadrupole model defined on the diamond lattice. The low‑energy degrees of freedom are represented by a planar XY vector mᵢ = (cos θᵢ, sin θᵢ) with fixed length, corresponding to the two electric quadrupole operators O₂₀ and O₂₂ of the non‑Kramers Γ₃ doublet. The Hamiltonian contains (i) bilinear exchange interactions up to fourth neighbours (J₁ = 1 as the energy unit, J₄ > 0 to stabilize ordering at the four symmetry‑related L‑points q = (π,π,π) etc.), (ii) a pseudo‑Zeeman coupling –h cos θᵢ that originates from the magnetic‑field‑induced mixing of excited crystal‑field levels (h ∝ |H|², positive for H∥