Electrons in solids have various degrees of freedom, such as charge, spin, and orbital, which are a source of fascinating electronic orderings in condensed matter physics. Atomic-scale multipoles have been introduced to provide a unified description of the mutual interplay between these degrees of freedom for electrons [1,2]. The concept of atomic-scale multipoles has been applied primarily to f-electron compounds, which exhibit many unconventional phase transitions. Among them, Pr-based compounds demonstrate multiple exotic phenomena associated with electronic multipoles, such as non-Fermi liquids, unconventional phase transitions, and superconductivity [3]. The key factor underlying these phenomena is the non-Kramers ground-state doublet (Γ₃non-Kramers doublet) for which the magnetic dipole is not activated; indeed, the transition to conventional magnetic orderings is prohibited. In addition, two Γ₃-type electric quadrupoles, namely O₂₀ (3z²-r²) and O₂₂ (x²-y²), are activated, which lead to unconventional multipole orderings that are not explained by simple charge and magnetic orderings.

Recently, Tsunetsugu et al. [4] showed a rich phase diagram consisting of the ferro- and antiferroelectric quadrupole orderings for Pr-based compounds. By analyzing a minimal model for an fcc lattice system [Fig. 1(a)] under the mean-field approximation, they obtained a variety of electric quadrupole orderings in the ground state [Fig. 1(b)]. The resultant quadrupole configurations are shown in Fig. 1(c). Interestingly, they also identified exotic partial orders of quadrupoles, which are denoted as zyox and zoxy, at finite temperatures, as shown in Fig. 1(d). These partial orders are characterized by the coexistence of the quadrupole order in the three-sublattice network and the disordered state in the remaining sublattice. Notably, they showed that the third-order term with respect to the ordered parameter in the free energy plays an important role in stabilizing the aforementioned partial quadrupole orders. This emergent third-order term in the free energy, which does not appear in the case of the conventional magnetic orderings, is characteristic of the electric quadrupole ordering. Meanwhile, this third-order term is relevant for the nematic phase in liquid crystals [5], which indicates that the obtained quadrupole orderings correspond to the nematic states of electronic origin. Their results are set to inspire future studies not only on relevant materials, such as PrMgNi₄ [6], but also further exploration of multipole-induced exotic electronic orderings.

Fig. 1 (a) Fcc lattice. (b) Ground-state phase diagram in the plane of the isotropic interaction, J, and the anisotropic interaction, K, expressed in units of the energy difference, E₁, between the Γ₃ and Γ₁ levels. (c) Schematics of the two- and four-sublattice quadrupole orderings. The yellow lines represent the symmetry axes 3z²-r², 3x²-r², and 3y²-r². (d) Finite-temperature (T) phase diagram. [These figures have been taken from Figs. 1(a), 2(c), 3, and 5(d) of Ref. 4].

References

[1] Y. Kuramoto, H. Kusunose, and A. Kiss, J. Phys. Soc. Jpn. 78, 072001 (2009).
[2] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Rev. Mod. Phys. 81, 807 (2009).
[3] T. Onimaru and H. Kusunose, J. Phys. Soc. Jpn. 85, 082002 (2016).
[4] H. Tsunetsugu, T. Ishitobi, and K. Hattori, J. Phys. Soc. Jpn. 90, 043701 (2021).
[5] P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).
[6] Y. Kusanose, T. Onimaru, G.-B. Park, Y. Yamane, K. Umeo, T. Takabatake, N. Kawata, and T. Mizuta, J. Phys. Soc. Jpn. 88, 083703 (2019).