Exploring rare-earth Kitaev magnets by massive-scale computational analysis

Abstract

The Kitaev honeycomb model plays a pivotal role in the quest for quantum spin liquids, in which fractional quasiparticles would provide applications in decoherence-free topological quantum computing. The key ingredient is the bond-dependent Ising-type interactions, dubbed the Kitaev interactions, which require strong entanglement between spin and orbital degrees of freedom. Here we investigate the identification and design of rare-earth materials displaying robust Kitaev interactions. We scrutinize all possible 4f electron configurations, which require up to 6+ million intermediate states in the perturbation processes, by developing a parallel computational program designed for massive-scale calculations. Our analysis reveals a predominant interplay between the isotropic Heisenberg J and anisotropic Kitaev K interactions across all realizations of the Kramers doublets. Remarkably, instances featuring 4f³ and 4f¹¹ configurations showcase the prevalence of K over J, presenting unexpected prospects for exploring the Kitaev quantum spin liquids in compounds, including Nd³⁺ and Er³⁺, respectively.

Introduction

In the realm of quantum spin liquids (QSLs), quantum fluctuations prevent localized magnetic moments from establishing conventional magnetic order, wherein excited nonlocal quasiparticles hold promise for applications in decoherence-free topological quantum computing^{1,2,3,4,5,6,7}. The Kitaev model, exhibiting exchange frustration between localized magnetic moments residing on a honeycomb lattice, is one viable model for the QSL since it is exactly solvable through the introduction of Majorana fermions³. The Hamiltonian is given by ${{{\mathcal{H}}}}_{{{\rm{Kitaev}}}}={\sum }_{\mu }{\sum }_{{\langle i,i^{\prime} \rangle }_{\mu }}K{S}_{i}^{\mu }{S}_{i^{\prime} }^{\mu }$, where K represents the coupling constant for the bond-dependent Ising-type interactions on three-type μ (x, y, and z) bonds on the honeycomb lattice, while ${S}_{i}^{\mu }$ (${S}_{i^{\prime} }^{\mu }$) signifies the μ component of the spin-1/2 operator at site i (its nearest-neighbor site $i^{\prime}$ on the μ bond). The realization of Kitaev-type interactions has been achieved in spin-orbit coupled Mott insulators, where the interplay of electron correlation and spin-orbit coupling (SOC) is pivotal^8,9. This is notably evident in the spin-orbital entangled Kramers doublet Γ₇, described by j_eff = 1/2 pseudospins, which typically originates from the low-spin d⁵ electron configuration under the octahedral crystal field (OCF). Indeed, the presence of dominant Kitaev interactions has been unveiled for several quasi-two-dimensional honeycomb compounds, such as A₂IrO₃ (A = Na, Li), α-RuCl₃, and other related materials^{9,10,11,12,13,14,15,16,17,18,19,20,21}. In these compounds, the Kitaev interactions stem from second-order perturbation processes with respect to the electron hopping mediated by ligands in edge-sharing MX₆ octahedra, where M represents a transition metal cation, and X denotes a ligand ion (see Fig. 1a).

Fig. 1: Challenge in the derivation of effective exchange interactions by second-order perturbation calculations for 4f-electron systems.

a Schematic of the Kitaev model realized in an edge-sharing network of RX₆ octahedra. Three-type μ (x, y, and z) bonds on the honeycomb lattice are distinguished. The hopping paths R-X-R on a z bond are represented by the purple lines. b Schematic of the calculations. We successfully cover the second-order perturbation calculations on a massive scale by employing a parallelization scheme spanning all the possible intermediate states $\left\vert 4{f}^{n-1}\right\rangle \otimes \left\vert 4{f}^{n+1}\right\rangle$, given the initial and final states $\left\vert 4{f}^{n}\right\rangle \otimes \left\vert 4{f}^{n}\right\rangle$. The sequence of perturbation processes is schematically depicted for the case of n = 5: 1 represents the initial state $\left\vert 4{f}^{5}\right\rangle \otimes \left\vert 4{f}^{5}\right\rangle$, 2 denotes an f-electron hopping from one $\left\vert 4{f}^{5}\right\rangle$ to the other $\left\vert 4{f}^{5}\right\rangle$, 3 represents an intermediate state $\left\vert 4{f}^{4}\right\rangle \otimes \left\vert 4{f}^{6}\right\rangle$, 4 denotes an f-electron hopping from $\left\vert 4{f}^{6}\right\rangle$ to $\left\vert 4{f}^{4}\right\rangle$, and 5 represents the final state $\left\vert 4{f}^{5}\right\rangle \otimes \left\vert 4{f}^{5}\right\rangle$ that is the same as the initial state. c The number of the intermediate states, N_int, for the 4fⁿ-4fⁿ states: N_int = 182 for n = 1 and n = 13, N_int = 182,182 for n = 3 and n = 11, and N_int = 6,012,006 for n = 5 and n = 9. The low-spin d⁵ case with N_int = 30 is shown for comparison. The initial/final and the intermediate states are schematically depicted for each electron configuration.

Besides the d-electron transition metal compounds, rare-earth materials with f electrons meet the requirements for the Kitaev interactions: cooperation of electron correlation and SOC. There exist rare-earth quasi-two-dimensional honeycomb materials, which would be deemed as intriguing platforms for realizing the Kitaev model, e.g, Na₂PrO₃^22,23, SmI₃²⁴, DyCl₃²⁵, ErX₃ (X = Cl, Br, I)^26,27, and YbCl₃²⁸. Notably, the expectation is that antiferromagnetic (AFM) Kitaev interactions would manifest in the 4f¹ electron configuration in A₂PrO₃ including Na₂PrO₃, in contrast to the ferromagnetic (FM) ones that typically dominate in d-electron systems^20,29,30. Nonetheless, the design and discovery of f-electron materials with strong Kitaev-type interactions remain elusive. This is mainly because the formulation of low-energy effective models for 4f-electron systems remains a formidable challenge, as it requires significant computational efforts for second-order perturbation calculations (see Fig. 1b). Indeed, the number of the intermediate states in the perturbation processes becomes 182,182 and 6,012,006 for 4f³ (4f¹¹) and 4f⁵ (4f⁹), respectively, in stark contrast to only 30 for the low-spin d⁵ electron configurations (see Fig. 1c).

To address this issue, we develop a highly parallel computational program capable of exhaustively performing second-order perturbation calculations on a massive scale. The program comprises three key steps. First, the eigenvectors $\left\vert 4{f}^{n}\right\rangle$ and the eigenvalues ${E}_{4{f}^{n}}$ for all the many-electron states with 4fⁿ electron configurations are prepared. In this step, the Coulomb interaction ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ between 4f electrons is taken into account, along with the subsequent SOC ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$, based on the Russell-Saunders coupling scheme³¹. Second, upon using these many-electron states, the initial and final 4fⁿ-4fⁿ states for neighboring sites (the tensor products $\left\vert 4{f}^{n}\right\rangle \otimes \left\vert 4{f}^{n}\right\rangle$ of 4fⁿ state pairs) and all the possible 4fⁿ⁻¹-4fⁿ⁺¹ intermediate states (the tensor products $\left\vert 4{f}^{n-1}\right\rangle \otimes \left\vert 4{f}^{n+1}\right\rangle$ of 4fⁿ⁻¹ and 4fⁿ⁺¹ state pairs) are automatically generated. Third, effective magnetic couplings are estimated based on the second-order perturbation expansion with respect to the 4f electron hopings ${{{\mathcal{H}}}}_{{{\rm{hop}}}}$. It is worth noting that the program can be flexibly extended beyond the second-order perturbation; it is capable of computing higher-order contributions, including multiple-spin interactions. We emphasize that even the second-order perturbation calculations are impracticable without efficient parallel computation (Fig. 1a) since the number of the intermediate states exceeds 6+ million for the 4f⁵ and 4f⁹ cases (Fig. 1b). This parallelization is achieved by implementing the Message Passing Interface in the C++ programming language.

In this study, we employ the program for the design of rare-earth Kitaev-type materials. For the 4fⁿ-4fⁿ states with n = 1, 3, 5, 9, 11, and 13, we assume a perfect OCF ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ within the edge-sharing RX₆ octahedra (R = rare-earth ions), along with ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ and ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$. This results in the formation of spin-orbital entangled Kramers doublet for all n, depending on the crystal field parameters. In the perturbation, we take into account the indirect 4f-p-4f electron hoppings ${{{\mathcal{H}}}}_{{{\rm{hop}}}}$ via p orbitals of ligand X with the use of the Slater-Koster transfer integrals t_pfπ and t_pfσ³², and the p-4f energy difference Δ_p-f in the intermediate states. Our analysis reveals that in all cases the low-energy Hamiltonian can be effectively described by two predominant exchange interactions between the pseudospins for the Kramers doublet: the bond-independent isotropic Heisenberg interaction denoted as J [given in ${{{\mathcal{H}}}}_{{{\rm{Heisenberg}}}}={\sum }_{\langle i,i^{\prime} \rangle }J{{{\bf{S}}}}_{i}\cdot {{{\bf{S}}}}_{i^{\prime} }$, where ${{{\bf{S}}}}_{i}={({S}_{i}^{x},{S}_{i}^{y},{S}_{i}^{z})}^{{{\rm{T}}}}$] and the bond-dependent anisotropic Kitaev interaction K. In most instances, both J and K exhibit AFM behavior. Notably, in the cases of 4f³ (as exemplified in Nd³⁺) and the electron-hole counterpart 4f¹¹ (Er³⁺), we find that K largely dominates over J, which realizes situations close to the pure Kitaev model. This finding opens up unexpected opportunities for investigating the Kitaev QSLs in 4f-electron systems. Furthermore, beyond the scope of the Kitaev model, our computational program can also be applied to a wide range of 4f-electron magnets, which would contribute to future exploration of exotic rare-earth magnetism.

Results

Ground-state Kramers doublets

Let us begin with the analysis of the crystal field splitting of the ground-state multiplets given by the Russell-Saunders coupling scheme, focusing on the 4fⁿ electron configurations with odd n²⁰. In 4f¹ electron configuration (Fig. 2a), the Coulomb interaction ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ is irrelevant, leaving 14-fold ²F manifold. This is split by ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ into the ²F_5/2 sextet and the ²F_7/2 octet. The ground-state ²F_5/2 sextet is further split by ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ into the Γ₇ doublet and Γ₈ quartet. Since the Γ₇ doublet has lower energy than the Γ₈ quartet, the 4f¹ case gives the Γ₇ Kramers doublet in the ground state. In the 4f³ electron configuration (Fig. 2b), ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ gives 52-fold ⁴I manifold in the ground state, which is split by ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ into four multiplets ⁴I_9/2, ⁴I_11/2, ⁴I_13/2, and ⁴I_15/2. The lowest-energy ⁴I_9/2 dectet is further split by ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ into the Γ₆ doublet and two Γ₈ quartets. The ground state depends on the crystal field parameters B₄₀ and B₆₀ (see Methods), and the Γ₆ Kramers doublet is selected when B₄₀ is predominant. In the 4f⁵ electron configuration (Fig. 2c), the lowest-energy ⁶H_5/2 sextet selected by ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ and ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ is split by ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ into the Γ₇ doublet and the Γ₈ quartet, and the ground state is given by the lower-energy Γ₇ Kramers doublet, similar to the 4f¹ case. In the 4f⁹ electron configuration (Fig. 2d), the lowest-energy ⁶H_15/2 sexdectet selected by ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ and ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ is split by ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ into the Γ₆ doublet, the Γ₇ doublet, and the three Γ₈ quartets. The ground state is either the Γ₆ doublet or the Γ₇ doublet depending on the crystal field parameters. In the 4f¹¹ electron configuration (Fig. 2e), the ground state is given by the Γ₇ doublet when B₆₀ is predominant. Finally, in the 4f¹³ electron configuration (Fig. 2f), the ground state is given by the Γ₆ doublet, irrespective of the crystal field parameters. Thus, all 4fⁿ cases considered here can offer the Kramers doublet in the ground state of an isolated ion.

Fig. 2: Schematic representation of multiplet splittings for 4fⁿ electron configurations with odd integers n (except for n = 7).

a 4f¹, b 4f³, c 4f⁵, d 4f⁹, e 4f¹¹, and f 4f¹³ cases are represented. In each configuration, the ground-state multiplet ${2S+1\atop }L$, initially determined by the Coulomb interaction ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ (left), undergoes splitting by the spin-orbit coupling ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$, resulting in the ground-state multiplet ${2S+1\atop }{L}_{J}$ (middle). Subsequently, ${2S+1\atop }{L}_{J}$ is further split by the octahedral crystal field ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$, leading to the formation of ground-state Kramers doublets Γ₇ (in red) or Γ₆ (in blue) or quartets Γ₆ (in thick green). In the cases of 4f³ in (b), 4f⁹ in (d), and 4f¹¹ in (e), the ground state is contingent upon the parameters B₄₀ and B₆₀ governing ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$; we present two extreme cases of B₄₀ = 0 (left) and B₆₀ = 0 (right). The 4f⁷ case is not shown as the orbital is quenched. The corresponding wave functions for the Kramers doublets are also depicted, with red and blue denoting spin-up and spin-down density profiles, respectively.

Table 1 explicitly enumerates all the accessible ground-state Kramers doublets characterized by the pseudospin j_eff = 1/2. In this table, each j_eff = 1/2 state is described with j and j^z representations; j (j^z) is the (secondary) total angular momentum quantum number. For the pseudospin j_eff = 1/2 degree of freedom, one can introduce the operator ${{\bf{S}}}={({S}^{x},{S}^{y},{S}^{z})}^{{{\rm{T}}}}$ defined by

$${S}^{\mu }={\mathbb{S}}\left(\begin{array}{cc}\left\langle +\right\vert {j}^{\mu }\left\vert +\right\rangle &\left\langle +\right\vert {j}^{\mu }\left\vert -\right\rangle \\ \left\langle -\right\vert {j}^{\mu }\left\vert +\right\rangle &\left\langle -\right\vert {j}^{\mu }\left\vert -\right\rangle \end{array}\right)=\frac{1}{2}{\sigma }^{\mu },$$

(1)

where j = (j^x, j^y, j^z) and σ = (σ^x, σ^y, σ^z) are the vectors of the total angular momentum operators and the Pauli matrices, respectively, and ${\mathbb{S}}$ is a real scalar.

Table 1 Ground-state multiplets and possible Kramers doublets for 4fⁿ electron configurations with odd integers n (except for n = 7)

Effective exchange couplings

Subsequently, the program proceeds to determine the second-quantized representations with multiple f-orbital bases for $\left\vert 4{f}^{n}\right\rangle \otimes \left\vert 4{f}^{n}\right\rangle$ by the aforementioned Kramers doublets, which is commonly used for the initial and final states of the perturbation. Additionally, it constructs the representations for all conceivable intermediate states $\left\vert 4{f}^{n-1}\right\rangle \otimes \left\vert 4{f}^{n+1}\right\rangle$. The energy difference between the initial/final states and the intermediate states is determined by two key parameters in the Hamiltonian ${{{\mathcal{H}}}}_{{{\rm{int}}}}$, namely, the onsite Coulomb interaction U and the Hund’s-rule coupling J_H, as well as another in ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$, namely, the SOC coefficient λ. For obtaining the values of U and λ, the Herbst-Wilkins table³³ and the Freeman-Watson table³⁴ are consulted, respectively. The parameter J_H is adjusted to achieve the alignment of energy differences between different multiplets according to the Dieke diagram³⁵; see Supplementary Note 1.

Given the representations and the excitation energies described above, J and K are calculated by employing the parallelization scheme for perturbation calculations spanning the intermediate states. For the hopping parameters, we adopt the Slater-Koster transfer integrals³², while changing the ratio ∣t_pfπ/t_pfσ∣ between 0 and 1; we take t_pfπ/t_pfσ < 0. The results are summarized in Fig. 3 for all 4fⁿ cases. In the 4f¹ Γ₇ case (Fig. 3a), which includes 182 4f⁰-4f² intermediate states, our results emphasize the dominance of the AFM K over compatibly the subdominant AFM J in the wide range of 0 < ∣t_pfπ/t_pfσ∣ ≤ 0.8. This behavior aligns with the findings based on the first-principles calculations in refs. ^30,31. The magnitudes of K and J both monotonically increase with ∣t_pfπ/t_pfσ∣, which is a general trend seen also for most of the other cases below. In the 4f³ Γ₆ case with 182,182 4f²-4f⁴ intermediate states (Fig. 3b), the intriguing scenario arises in which the AFM K overwhelmingly outweighs non-negligible AFM J; this is particularly pronounced at ∣t_pfπ/t_pfσ∣ ≃ 1.0, where J almost vanishes. We also emphasize that K is one order of magnitude larger than that in the 4f¹ case. Note that the coupling constants K and J are given in unit of ${t}_{pf\sigma }^{4}{\Delta }_{p-f}^{-2}$ eV⁻¹; we will discuss the actual values later. In the 4f⁵ Γ₇ case with 6,012,006 4f⁴-4f⁶ intermediate states (Fig. 3c), the AFM K becomes predominant compared to the subdominant AFM J in the entire region of ∣t_pfπ/t_pfσ∣. In the 4f⁹ case (Fig. 3d), there are two cases, Γ₆ and Γ₇, depending on the crystal field parameters, both of which include 6,012,006 4f⁸-4f¹⁰ intermediate states. In the Γ₆ case, K turns to be FM, while J remains AFM and predominant compared to K. Meanwhile, in the Γ₇ case, both K and J are AFM, while J is again predominant. In the case of 4f¹¹ Γ₇ (Fig. 3e), the trends mirror the electron-hole counterpart, the 4f³ Γ₆ case; the AFM K becomes far predominant compared to the AFM J, while J does not decrease for large ∣t_pfπ/t_pfσ∣. We note that the magnitude of K is also large comparable to the 4f³ case. Finally, in the 4f¹³ Γ₆ case (Fig. 3f), the result is quite different from the electron-hole counterpart, the 4f¹ case; K is notably suppressed compared to the predominant AFM J. This is, however, consistent with the prior findings in ref. ³⁶. It should be noted that the off-diagonal terms, referred to as Γ and Γ′, were found to be zero across all the cases since we omit the direct 4f-4f electron hoppings in the present analyses.

Fig. 3: Two coupling constants, isotropic Heisenberg interaction J and anisotropic Kitaev interaction K, derived by the second-order perturbation for 4fⁿ-4fⁿ electron configurations with odd integers n (except for n = 7).

a 4f¹ Γ₇, b 4f³ Γ₆, c 4f⁵ Γ₇, d 4f⁹ Γ₆ and Γ₇, e 4f¹¹ Γ₇, and f 4f¹³ Γ₆ cases are represented. The data are plotted for ∣t_pfπ/t_pfσ∣; we take t_pfπ/t_pfσ < 0.

The results in Fig. 3 highlight that, in the majority of instances, aside from the 4f⁹ and 4f¹³ cases, the AFM K prevails over the subdominant AFM J. This suggests a heightened propensity for robust Kitaev interactions within diverse 4f-electron systems. This is more clearly demonstrated by plotting the ratio of ∣K/J∣ in Fig. 4. Except for the 4f⁹ and 4f¹³ cases (and the 4f¹ case for large ∣t_pfπ/t_pfσ∣), ∣K/J∣ is greater than 1, indicating the predominant Kitaev interactions. Interestingly, besides the 4f¹ case, ∣K/J∣ consistently exhibits monotonic increases with ∣t_pfπ/t_pfσ∣. It is noteworthy that the substantial predominance of AFM K over AFM J is particularly viable, especially in the cases of 4f³ Γ₆ and 4f¹¹ Γ₇. In both scenarios, it is observed that ∣K/J∣ > 4 for ∣t_pfπ/t_pfσ∣ ≳ 0.6, which includes the realistic range of the parameters³⁷. In addition, the magnitude of K is considerably larger than in the other cases. To emphasize these prominent properties, we show the estimates of J, K, and ∣K/J∣ in Table 2, assuming the typical values of the parameters as t_pfσ = 0.35 eV, t_pfπ/t_pfσ = −0.7³⁷, and Δ_p-f = 1 eV. Notably, for the 4f³ Γ₆ and 4f¹¹ Γ₇ configurations, it is demonstrated that K = 1.21 meV and 1.27 meV, respectively, which are one order of magnitude larger than the other cases, and furthermore, K/J = 6.89 and 5.21, signifying the substantial AFM K prevalence over the AFM J.

Fig. 4: Ratio between the Kitaev and Heisenberg interactions, ∣K/J∣, for different 4fⁿ cases.

We exclude the data when ∣K∣ or ∣J∣ are extremely small: $| K| \, < \, 1{0}^{-5}{t}_{pf\sigma }^{4}{\Delta }_{p{{-}}f}^{-2}$ eV⁻¹ and $| J| \, < \, 1{0}^{-5}{t}_{pf\sigma }^{4}{\Delta }_{p{{-}}f}^{-2}$ eV⁻¹.

Table 2 Plausible estimates of the isotropic Heisenberg interaction J, anisotropic Kitaev interaction K, and their ratio ∣K/J∣ for various Kramers doublets in 4f-electron systems

Candidate materials

Let us finally discuss candidate materials for the 4f¹, 4f³, 4f⁵, and 4f¹¹ cases where K dominates J in our calculations. First, for 4f¹, the authors and their collaborators previously identified A₂PrO₃ (A = alkali metals) as potential Kitaev-type magnets with the conventional assumption in the Russell-Saunders coupling scheme whereby the ordering of energy scales is given as ${{{\mathcal{H}}}}_{{{\rm{int}}}} \, > \, {{{\mathcal{H}}}}_{{{\rm{SOC}}}} \, \gg \, {{{\mathcal{H}}}}_{{{\rm{OCF}}}}$^29,30. However, the tetravalent Pr⁴⁺ ion is recently recognized to reside in the intermediate coupling regime ${{{\mathcal{H}}}}_{{{\rm{SOC}}}} \sim {{{\mathcal{H}}}}_{{{\rm{OCF}}}}$^38,39,40. We have verified that in this regime the AFM K is reduced to be subdominant, while the AFM J prevails⁴¹. Second, for 4f³, Nd³⁺-based materials are considered promising, although honeycomb lattice compounds with Nd³⁺ have not yet been identified to the best of our knowledge. This observation suggests avenues for additional materials design in the exploration of Nd³⁺-based Kitaev-type magnets. Third, for 4f⁵, SmI₃ has recently undergone experimental scrutiny as a potential host for the Kitaev QSL, given its absence of long-range spin magnetic order down to 0.1 K²⁴. Further experiments are awaited to identify the relevant magnetic interactions. Finally, for 4f¹¹, Er³⁺-based van der Waals magnets ErX₃ (X = Cl, Br, I) were studied^26,27. These materials have similar lattice structures to that of the prime candidate for the Kitaev QSL, α-RuCl₃, and were shown to exhibit noncollinear vortex-type magnetic orders. A recent experiment for ErBr₃ discussed the relevance of long-range dipolar interactions⁴². However, note that similar vortex-like magnetic orders were also found in extensions of the Kitaev mdoel^30,43,44. It would be intriguing to revisit the Er³⁺-based materials by using ab initio approaches.

Discussion

Our comprehensive approach, leveraging a parallel computational program capable of massive-scale second-order perturbation calculations, has provided insights into the nature of exchange interactions in rare-earth quasi-two-dimensional honeycomb lattices. The observed dominance of the anisotropic Kitaev interaction over the isotropic Heisenberg interaction in certain cases, particularly for 4f³ and 4f¹¹ configurations, opens new avenues for investigating the Kitaev-type QSL. In particular, our results highlight Nd³⁺ and Er³⁺-based magnets as plausible candidates for the Kitaev QSL. The developed computational program extends its utility beyond the Kitaev model, which would address a wide range of exchange interactions in 4f-electron systems. Particularly when used in conjunction with ab initio calculations, which allow for an in-depth numerical analysis of parameters for crystal fields and electron hoppings, the program will more accurately describe spin systems in 4f-electron systems, including the effects of not only direct 4f-4f direct hopping but also charge-transfer processes and cyclic exchanges^14,45, which will be explored in subsequent researches. We also note that, although we demonstrated the simplest cases of OCF in this study, the program is applicable to any perturbation problem for various types of crystal fields with lower symmetry. This work not only contributes to advancing our understanding of rare-earth Kitaev-type materials but also lays the groundwork for future exploration of exotic magnetism in this intriguing field of research.

We note that computational speed could be significantly enhanced by separating ${c}_{im\sigma }^{{\dagger} }$ and ${c}_{i^{\prime} m^{\prime} \sigma }$ in ${{{\mathcal{H}}}}_{{{\rm{hop}}}}$⁴⁶, given that the site indices i and $i^{\prime}$ pertain exclusively to subspaces with 4fⁿ⁺¹ and 4fⁿ⁻¹ configurations, respectively [see equations (12) and (13)]. This strategy would reduce the size of the perturbation calculations from $\scriptstyle2\left({14}\atop{n-1}\right)\left({14}\atop{n+1}\right)$ (considering $\left\vert 4{f}^{n-1}\right\rangle \otimes \left\vert 4{f}^{n+1}\right\rangle$) to $\scriptstyle14\left(\left({14}\atop{n-1}\right)+\left({14}\atop{n+1}\right)\right)$ (considering $\left\vert 4{f}^{n-1}\right\rangle$ and $\left\vert 4{f}^{n+1}\right\rangle$ separately). For example, at n = 6, this approach would decrease the size from 6,012,066 to 76,076, demonstrating a substantial reduction that warrants testing in future studies.

Methods

Coulomb interactions

The Hamiltonian ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ describing the Coulomb interactions between f electrons is given by

$$\begin{array}{l}{{{\mathcal{H}}}}_{{{\rm{int}}}}={\sum} _{i}\sum\limits_{{m}_{1},{m}_{2},{m}_{3},{m}_{4}}\sum\limits_{{\sigma }_{1},{\sigma }_{2}}{\delta }_{{m}_{1}+{m}_{2},{m}_{3}+{m}_{4}}\sum\limits_{k=0,2,4,6}\\ {F}^{k}{C}^{(k)}({m}_{1},{m}_{4}){C}^{(k)}({m}_{2},{m}_{3}){c}_{i{m}_{1}{\sigma }_{1}}^{{\dagger} }{c}_{i{m}_{2}{\sigma }_{2}}^{{\dagger} }{c}_{i{m}_{3}{\sigma }_{2}}{c}_{i{m}_{4}{\sigma }_{1}},\end{array}$$

(2)

where F^k and C^(k) denote the Slater-Condon parameters and the Guant coefficients, respectively (k = 0, 2, 4, 6); δ is the Kronecker delta; ${c}_{im\sigma }^{{\dagger} }$ and c_imσ represent creation and annihilation operators of an electron at site i in the spherical harmonics basis, respectively (m and σ = ±1 denote the magnetic and spin quantum numbers, respectively). Here, the Slater-Condon parameters are related with the onsite Coulomb interaction U and the Hund’s-rule coupling J_H as^47,48

$$U={F}^{0},$$

(3)

$${J}_{{{\rm{H}}}}=\frac{1}{6435}\left(286{F}^{2}+195{F}^{4}+250{F}^{6}\right).$$

(4)

${{{\mathcal{H}}}}_{{{\rm{int}}}}$ is diagonalized using the lowering operators of orbital and spin angular momenta, L⁻ and S⁻, respectively, both of which commute with ${{{\mathcal{H}}}}_{{{\rm{int}}}}$:

$${L}^{-}={\sum}_{i}{\sum}_{{m}_{1},{m}_{2}}{\sum}_{\sigma }{\delta }_{{m}_{1},{m}_{2}+1}\sqrt{(\ell +m+1)(\ell -m)}{c}_{i{m}_{1}\sigma }^{{\dagger} }{c}_{i{m}_{2}\sigma },$$

(5)

where ℓ is the orbital quantum number taken as ℓ = 3 for the f-orbital manifold, and

$${S}^{-}={\sum}_{i}{\sum}_{m}{\sum}_{{\sigma }_{1},{\sigma }_{2}}{\delta }_{{\sigma }_{1},{\sigma }_{2}+1}\sqrt{\left(\frac{1}{2}+\frac{{\sigma }_{2}}{2}+1\right)\left(\frac{1}{2}-\frac{{\sigma }_{2}}{2}\right)}{c}_{im{\sigma }_{1}}^{{\dagger} }{c}_{im{\sigma }_{2}}.$$

(6)

A leading eigenvector of the ground-state multiplet ${}^{2{S}_{1}+1}{L}_{1}$ with L₁ (S₁) being the largest L (S) for 4fⁿ electron configuration is given as $\underbrace{{c}_{i,\ell ,+1}^{{\dagger} }\cdots {c}_{i,\ell -n+1,+1}^{{\dagger} }}_{n}\left\vert 0\right\rangle$, where $\left\vert 0\right\rangle$ denotes the vacuum state. Given this leading eigenvector, all the eigenvectors expanded within the ${}^{2{S}_{1}+1}{L}_{1}$ subspace can be derived by successively applying either L⁻ or S⁻ to the leading eigenvector. Then, the leading eigenvector for the multiplet ${}^{2{S}_{1}+1}{L}_{2}$, where L₂ is the second-largest L, is constructed as a vector orthogonal to the eigenvector within the ${}^{2{S}_{1}+1}{L}_{1}$ subspace with the expectation values of L^z and S^z being L₂ and S₁, respectively:

$${L}^{z}={\sum}_{i}{\sum}_{m}{\sum}_{\sigma }m{c}_{im\sigma }^{{\dagger} }{c}_{im\sigma },$$

(7)

$${S}^{z}={\sum}_{i}{\sum}_{m}{\sum}_{\sigma }\frac{\sigma }{2}{c}_{im\sigma }^{{\dagger} }{c}_{im\sigma }.$$

(8)

Similarly, all the eigenvectors within the ${}^{2{S}_{1}+1}{L}_{2}$ subspace can be derived by successively applying either L⁻ or S⁻ to the leading eigenvector. This process, involving the application of either L⁻ or S⁻ to the leading eigenvector with orthogonality, is repeated for the remaining subspaces as well. Finally, we assess the numerical validity of the derived eigenvectors by examining their orthogonality and verifying the absence of nonzero off-diagonal elements in ${{{\mathcal{H}}}}_{{{\rm{int}}}}$.

Spin-orbit coupling

The Hamiltonian ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ describing the effect of the SOC is given by

$${{{\mathcal{H}}}}_{{{\rm{SOC}}}}={\sum}_{i}{{{\mathcal{H}}}}_{{{\rm{SOC}}},i},$$

(9)

where

$$\begin{array}{l}{{{\mathcal{H}}}}_{{{\rm{SOC}}},i}=\frac{\lambda }{2}{\sum}_{m=-\ell }^{\ell }\sum\limits_{\sigma }m\sigma {c}_{im\sigma }^{{\dagger} }{c}_{im\sigma }\\ +\frac{\lambda }{2}{\sum }_{m=-\ell }^{\ell -1}\sqrt{\ell +m+1}\sqrt{\ell -m}({c}_{im+1-}^{{\dagger} }{c}_{im+}+{c}_{im+}^{{\dagger} }{c}_{im+1-}),\end{array}$$

(10)

where λ > 0 is the SOC coefficient.

${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ is diagonalized using the lowering operator of total angular momentum, J⁻, which commutes with ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$: J⁻ = L⁻ + S⁻. A leading eigenvector of the ${2S+1\atop }{L}_{{J}_{1}}$ with J₁ being the largest J (= L + S) is given as the eigenvector with the expectation value of J^z (= L^z + S^z) being J₁. All the eigenvectors within the ${2S+1\atop }{L}_{{J}_{1}}$ subspace can be derived by successively applying either J⁻ to the leading eigenvector. Then, the leading eigenvector of the ${2S+1\atop }{L}_{{J}_{2}}$, where J₂ is the second-largest J₂ (= L + S − 1), is constructed as a vector orthogonal to the eigenvector within the ${2S+1\atop }{L}_{{J}_{2}}$ subspace with the expectation values of J^z being J₂. This process, involving the application of J⁻ to the leading eigenvector with orthogonality, is repeated for the remaining subspaces as well. Finally, we assess the numerical validity of the derived eigenvectors by examining their orthogonality and verifying the absence of nonzero off-diagonal elements in ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$. We also confirm that this approach using L⁻ and S⁻ for ${{{\mathcal{H}}}}_{{{\rm{int}}}}$ and J⁻ for ${{{\mathcal{H}}}}_{{{\rm{SOC}}}}$ yields the same eigenvectors and eigenvalues for the 4f² electron configuration as those obtained using the Wigner 3-j symbols³⁰.

Octahedral crystal field

The Hamiltonian ${{{\mathcal{H}}}}_{{{\rm{OCF}}}}$ describing the octahedral crystal field is given by

$${{{\mathcal{H}}}}_{{{\rm{OCF}}}}={B}_{40}{O}_{4}+{B}_{60}{O}_{6},$$

(11)

where O₄ = O₄₀ + 5O₄₄ and O₆ = O₆₀ − 21O₆₄. O_rs (s = −r, −r + 1, ⋯, r) are the rank-r Stevens operators⁴⁹, and B₄₀ and B₆₀ are the coefficients.

Electron hopping

The Hamiltonian ${{{\mathcal{H}}}}_{{{\rm{hop}}}}$ describing the kinetic energy of electron hopping via indirect 4f-p-4f hopping processes is given by

$${{{\mathcal{H}}}}_{{{\rm{hop}}}}={\sum}_{\mu }{\sum}_{{\langle i,i^{\prime} \rangle }_{\mu }}{{{\mathcal{H}}}}_{{{\rm{hop}}},ii^{\prime} }^{\left(\mu \right)},$$

(12)

where ${{{\mathcal{H}}}}_{{{\rm{hop}}},ii^{\prime} }^{\left(\mu \right)}$ denotes the electron hopping between nearest-neighbor sites i and $i^{\prime}$ on the μ bond (μ = x, y, and z) as

$${{{\mathcal{H}}}}_{{{\rm{hop}}},ii^{\prime} }^{\left(\mu \right)}=\sum\limits_{m,m^{\prime} }\sum\limits_{\sigma =\pm }\left({\sum}_{o,p}\frac{{t}_{im,op,\sigma }{t}_{i^{\prime} m^{\prime} ,op,\sigma }}{{\Delta }_{p{{-}}f}}{c}_{im\sigma }^{{\dagger} }{c}_{i^{\prime} m^{\prime} \sigma }+{{\rm{h.c.}}}\right).$$

(13)

t_iu,op,σ is the transfer integral for spin σ between 4f orbital u at site i and p orbital p (= x, y, and z) at one of two ligand sites o (= 1 and 2) shared by two RX₆ octahedra for the sites i and $i^{\prime}$, and Δ_p-f is the energy difference between p and 4f orbitals. For t_im,op,σ and ${t}_{i^{\prime} m^{\prime} ,op,\sigma }$, we refer to ref. ³³.

Perturbation expansion

The effective Hamiltonian for a pair of j_eff = 1/2 pseudospins for nearest-neighbor sites i and $i^{\prime}$ on a μ bond is calculated by

$${h}_{ii^{\prime} }^{\left(\mu \right)}=\sum\limits_{a,b,c,d=\pm }{\sum}_{n}\frac{\left\langle c,d\right\vert {{{\mathcal{H}}}}_{{{\rm{hop}}},ii^{\prime} }^{(\mu )}\left\vert n\right\rangle \left\langle n\right\vert {{{\mathcal{H}}}}_{{{\rm{hop}}},ii^{\prime} }^{\left(\mu \right)}\left\vert a,b\right\rangle }{{E}_{0}-{E}_{n}}\left\vert c,d\right\rangle \left\langle a,b\right\vert .$$

(14)

where $\left\vert a,b\right\rangle$ and $\left\vert c,d\right\rangle$ are the initial and final two-site states with 4fⁿ-4fⁿ electron configurations described in Table 1 at each site, and $\left\vert n\right\rangle$ is the intermediate states with 4fⁿ⁺¹-4fⁿ⁻¹ electron configurations; E₀ is the energy for the initial and final states, while E_n is for the intermediate state $\left\vert n\right\rangle$.