Multipolar fluctuations in localized 4f squared-electron systems from dynamical mean-field theory: application to PrCdNi 4

The gist Multipolar-fluctuation analysis based on density functional theory combined with dynamical mean-field theory is extended from 4f– to 4f2–electron systems, applying this framework to PrCdNi4 to propose a (3z2 −r2)type antiferroquadrupolar order at q = (2π, π, 0) driven by competition between nearest- and fourthnearest-neighbor interactions.

Methodological Developments

The paper presents two methodological developments for extending multipolar fluctuation analysis to 4f2–electron systems. First, many-particle states are constructed within the j–j coupling scheme in the j = 5/2 subspace to reproduce the crystalline-electric-field splitting of the 4f2 Hund’s-rule ground-state multiplet. Second, multipolar susceptibilities and interactions between 4f2 multiplets are derived through a mapping from single-particle to many-particle bases. This framework is applied to PrCdNi4, which has a non-Kramers Γ3 doublet ground state, and proposes a (3z2 −r2)type antiferroquadrupolar order at q = (2π, π, 0) driven by the competition between nearest- and fourthnearest-neighbor interactions. The formalism developed here is applicable to general 4fn–electron systems.

CEF Levels within the j–j Coupling

The CEF levels of the J = 4 manifold of the 4f2 configuration are obtained by diagonalizing a many-body Hamiltonian constructed within the j–j coupling scheme in the j = 5/2 subspace, neglecting high-energy states. The CEF levels are calculated using a many-body Hamiltonian H given by Eq. (3), which includes spin-orbit coupling (HSOC), the CEF potential H(f1), and the f–f Coulomb interaction Hint. When restricting to the j = 5/2 subspace, an effective Hamiltonian H(5/2)eff is defined which includes both the CEF potential and Coulomb interaction terms. This effective Hamiltonian reproduces the LS-coupling results in Fig. 1(a) when considering only the zeroth-order terms corresponding to conventional j–j coupling. The first-order correction in 1/λ to the CEF potential, B˜(1) in Eq. (9), is given by a formula involving matrix elements of H(f1)-CEF and the spin-orbit coupling constant λ. This correction allows the non-Kramers Γ3 doublet to become the ground state.

Many-Body Susceptibility and Mapping

The conventional static susceptibility in the single-particle basis is given by Eq. (13), which can be regarded as an N2m × N2m matrix. The susceptibility expressed in the many-particle basis M is given by Eq. (14), forming an N2M × N2M matrix. The required susceptibility XM1M2M3M4 can be related to the conventional one χm1m2m3m4 via a transformation involving the density operator matrix element A. This leads to the forward transformation in Eq. (17) which expresses χm1m2m3m4 in terms of XM1M2M3M4. The inverse transformation is given by Eq. (18), which constructs X˜ = A+x as the solution to the L2-norm minimization problem.

Application to PrCdNi4

In PrCdNi4, DFT calculations and DMFT are used, treating 4f electrons as itinerant and then constructing a low-energy effective model using symmetry-protected maximally projected Wannier functions. The strong correlations are treated within DMFT using the effective j = 5/2 model. The CEF parameters w and x are adjusted to reproduce experimental CEF levels, yielding w = 4.23 K and x = 0.62, which reproduces the experimental value ∆Γ3−Γ4 = 13.8 K. The multipolar degrees of freedom are analyzed using symmetry analysis of possible fluctuation modes, focusing on operators active within the CEF ground-state doublet Γ3 and those appearing through mixing between Γ3 and the first-excited triplet Γ4.

Multipolar Susceptibilities and Order

The susceptibility calculations are performed using a strong-coupling limit (SCL) formula of Refs. [12, 16]. The quantities are transformed into the many-particle basis Xˆ, Xloc, and I(q). The leading instability occurs in the Qu channel at q = qW, indicating a Qu-type antiferroquadrupolar order with q = qW. The subleading fourth-NN interaction is essential for stabilizing the W-point order over the X-point order favored by the NN interaction. The transition temperature is estimated to be 2.0 K, which reproduces the correct order of magnitude for TO ≈ 1.0 K.

Conclusion

The present method enables the DFT+DMFT analysis of multipolar fluctuations for general 4fn configurations. The paper proposes a Qu-type quadrupolar order at the W point with the translation vector qW = (2π, π, 0). The subleading fourth-NN interaction is essential for stabilizing the W-point order over the X-point order favored by the NN interaction. The present method enables the DFT+DMFT analysis of multipolar fluctuations for general 4fn configurations. The paper applies this framework to PrCdNi4, which exhibits a phase transition at TO = 1.0 K within the Γ3 CEF doublet. The central challenge lies in the difference between the bases: The Hund’s-rule ground state consists of many-particle states, whereas the Green’s-function approach relies on a single-particle basis.

Applications to Other Systems

The present method enables the DFT+DMFT analysis of multipolar fluctuations for general 4fn configurations. Applications to other 4fn–electron systems, such as Nd and Sm compounds, are thereby made possible.

Acknowledgments

We thank T. Onimaru and H. Kusunose for valuable discussions. This work was supported by JSPS KAKENHI Grants No. 23H04869, No. 25K22013, and No. 26K00659. Part of the computations in this work were performed using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo (No. 2025-Ca-0086, No. 2025-Cb-0049, and 2026-Ca-0142. The paper is available as arXiv:cond-mat.str–el [cond-mat.str–el] 8 Oct 2026.

Appendix A and B Details

Appendix A details the calculation of matrix elements Amm′,MM′ defined in Eq. (16), which involves coupling between LS and jM states<ref:2610.

1. Bold header: Many-particle state construction for 4f2 systems

This methodology allows for many-particle states are constructed within the j–j coupling scheme in the j = 5/2 subspace, so as to reproduce the crystalline-electric-field splitting of the 4f2 Hund’s-rule ground-state multiplet. This capability enables AI systems to accurately model and predict energy level schemes for non-Kramers doublets without relying solely on approximations derived from single-particle states.

2. Bold header: Mapping of susceptibilities between bases

The development of a mapping from the single-particle susceptibility to the many-body susceptibility allows for direct analysis of multipolar fluctuations in the Hund’s rule ground multiplet. This enables AI systems to move beyond conventional single-particle representations to analyze collective, many-body phenomena like quadrupolar fluctuations directly.

3. Bold header: First-principles prediction of multipolar order

The framework permits the application of DFT+DMFT to first-principles exploration of multipolar physics in 4f n-electron systems. This allows AI systems to predict the existence and nature of orders, such as the proposed (3z2 −r2) -type antiferroquadrupolar order at q = (2π, π, 0), from first principles.

4. Bold header: Determination of effective CEF parameters

The procedure for adjusting CEF parameters w and x to reproduce the experimentally estimated 4f2(J = 4) CEF levels allows AI systems to bridge theoretical calculations with experimental data by systematically tuning model inputs until they match observed energy splittings, as shown in Appendix C.

5. Bold header: Momentum-dependent susceptibility analysis

By converting conventional static susceptibility in the single-particle basis to the many-particle basis using Eq. (17), AI systems can compute the momentum-dependent susceptibility XM1M2,M3M4, which is crucial for detecting phase transitions into multipolar ordered states.

6. Bold header: Characterization of fluctuation modes

The procedure for diagonalizing the transformed susceptibility matrix allows for the extraction of eigenmodes where the eigenvectors u(ξ) are projected onto the multipole basis Zˆγ, yielding the expansion coefficients c(ξ)γ, enabling classification by irreducible representation to identify dominant fluctuations.

7. Bold header: Real-space interaction modeling

The derivation of effective Heisenberg-type Hamiltonian H = −1/2 X ij,γ=u,v Q γ(ri)I γ(ri − rj)Q γ(rj) allows AI systems to construct effective Hamiltonians that incorporate both nearest-neighbor and fourth-nearest-neighbor interactions as a function of intersite distance.

Multipolar-fluctuation analysis based on density functional theory combined with dynamical mean-field theory is extended from 4f 1 - to 4f squared-electron systems. Two methodological developments are presented. First, many-particle states are constructed within the j - j coupling scheme in the j=5/2 subspace, so as to reproduce the crystalline-electric-field splitting of the 4f squared Hund's-rule ground-state multiplet. Second, multipolar susceptibilities and interactions between 4f squared multiplets are derived through a mapping from single-particle to many-particle bases. We apply this framework to PrCdNi 4, which has a non-Kramers Γ 3 doublet ground state, and propose a (3z 2-r 2) -type antiferroquadrupolar order at =(2π,π,0) driven by the competition between nearest- and fourth-nearest-neighbor interactions. The formalism developed here is applicable to general 4f n-electron systems.