Octupolar ordering ofΓ8ions in a magnetic field

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We study f-electron lattice models which are capable of supporting octupolar, as well as dipolar and quadrupolar, order. Analyzing the properties of the ${\ensuremath{\Gamma}}_{8}$ ground-state quartet, we find that (111)-type combinations of the ${\ensuremath{\Gamma}}_{5}$ octupoles ${\mathcal{T}}_{111}^{\ensuremath{\beta}}={\mathcal{T}}_{x}^{\ensuremath{\beta}}+{\mathcal{T}}_{y}^{\ensuremath{\beta}}+{\mathcal{T}}_{z}^{\ensuremath{\beta}}$ are the best candidates for octupolar order parameters. Octupolar ordering induces ${\ensuremath{\Gamma}}_{5}$-type quadrupoles as secondary order parameter. Octupolar order is to some extent assisted, but in its basic nature unchanged, by allowing for the presence of quadrupolar interactions. In the absence of an external magnetic field, equivalent results hold for antiferro-octupolar ordering on the fcc lattice. In this sense, the choice of our model is motivated by the recent suggestion of octupolar ordering in ${\mathrm{NpO}}_{2}.$ The bulk of our paper is devoted to a study of the effect of an external magnetic field on ferro-octupolar ordering. We found that octupolar order survives up to a critical magnetic field if the field is lying in specific directions, while for general field directions, the underlying symmetry of the model is destroyed and therefore the phase transition suppressed even in weak fields. Field-induced multipoles and field-induced couplings between various order parameters are discussed on the basis of a group-theoretical analysis of the Helmholtz potential. We also studied the effect of octupolar ordering on the nonlinear magnetic susceptibility which satisfies Ehrenfest-type relations at continuous octupolar transitions.