The heavy-fermion superconductor ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ exhibits a nonsuperconducting phase transition that precedes the emergence of superconductivity. The nature of the corresponding order parameter remains under debate, with competing proposals involving magnetic dipoles or electric quadrupoles. We derive the momentum-dependent multipolar susceptibilities and effective interactions among the localized $4f$ electrons, based on the framework of density functional theory combined with dynamical mean-field theory. Magnetic fluctuations within the crystalline-electric-field (CEF) ground-state doublet are dominated by $\mathit{q}=(1/2,1/2,0)$, corresponding to a two-dimensional checkerboard configuration of the magnetic moment $M_z$ along the $c$ axis. Hybridization between the CEF ground-state and the first-excited doublet gives rise to leading magnetic octupole fluctuations of $z(x^2-y^2)$ symmetry, followed by electric quadrupole fluctuations of $x^2-y^2$ and $\{yz,zx\}$ symmetries. By taking into account the anisotropic magnetic-field dependence of the transition temperature $T_0$, we conclude that an antiferromagnetic order of $M_z$ at $\mathit{q}=(1/2,1/2,0)$ is consistent with the experiments, owing to the enhancement of $T_0$ caused by fluctuations of the field-induced quadrupole of $\{yz,zx\}$ type under an in-plane magnetic field.

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• Magnetic order
• Multipole order
• Rare-earth magnetic materials
• Dynamical mean field theory