Introduction

The behavior of U-5f electrons in uranium-based compounds, extending from localization to mobility and dualism1,2,3, is further complicated by Kondo screening by conduction electrons, leading to the advent of phenomena in a variety of Kondo lattices. Amorese et al.3 elucidated the dual nature of local atomic-like multiplet states in Pauli-paramagnetic UFe2Si2, which has been known to exhibit Kondo-like behavior at low temperatures4. USbTe, a Kondo lattice ferromagnet, has been found to possess a large anomalous Hall conductivity, which is suggested to be due to the intrinsic Berry curvature hosted by the Kondo hybridization between the local magnetic moment of U-5f and the conduction electrons5. UTe2, a heavy fermion superconductor, has been reported to lack long-range magnetic ordering6, in contrast to other uranium compounds, such as USbTe and USbSe (TC ~ 127 K)7. Nonetheless, UTe2 has been found to exhibit pressure-induced long-range magnetic ordering8,9,10,11. Li et al.9 demonstrated that increasing pressure leads to a diminution of magnetic moments and long-range magnetic ordering. The manifestly disparate behaviors exhibited in U-5f systems reflect a profound bond between U-5f and its surroundings, and are not easily explicable by existing theories.

Theoretical studies have proposed the presence of orbital-selective Mott phases (OSMP) to explain the duality observed in multi-band correlated systems12,13,14. This phenomenon is characterized by the coexistence of localized electrons in certain orbitals and itinerant electrons in other orbitals. The OSMP has been employed to explain the dual nature of Fe-based compounds. For example, Kim et al.15 reported that the Mott transition in FePS3 can be orbital selective, with the t2g states undergoing a correlation-induced insulator-to-metal transition while the eg states remain gaped under pressure. However, no evidence of the occurrence of the OSMP has been found in the 5f system.

Unlike 4f electrons in the lanthanide elements, which are typically localized within the Mott physics16, the degree of localization of 5f electrons in the actinide elements is strongly dependent on the crystal structure1, crystal electric field, and spin-orbit coupling (SOC). These factors can affect the hybridization channels, leading to a varying screening of the U-5f magnetic moments17, thereby altering the magnetization mechanism depending on whether Kondo hybridization is coherent or incoherent. According to the multichannel Kondo model18,19, Kondo systems can be classified into three types (under, fully, over-screened) based on the local magnetic moment raised by impurity spin S and the number of conduction electron channels, n. In the under-screened case (n < 2S), S is partially screened at low temperatures, potentially allowing for long-range magnetic ordering due to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. An example of this is the Anderson lattice model of uranium monochalcogenides, where UTe is modeled with S = 1 and n < 220,21. However, the RKKY mechanism may not be applicable to coherent U-5f Kondo lattices, as the Kondo coupling and the RKKY interaction favor different ground states22,23.

In a Bloch electrons system, van Hove singularity (VHS)24 can be attributed as a source of magnetization. As the Fermi energy approaches a VHS of the electronic density of states (DOS), the DOS diverges, allowing for weak interactions to have a significant influence on the electronic behavior. This can result in instabilities in charge and spin susceptibilities, leading to substantial enhancement of ferromagnetism25,26 and antiferromagnetism27. The VHS has been associated with the local magnetic moment in a transition metal28 and graphene bilayer29. By using density functional theory (DFT) combined with dynamical mean field theory (DMFT), Hausoel et al.28 recently investigated the paramagnetic phase spectrum of Ni metal. Their results revealed a van Hove magnet at the L point of the Brillouin zone, characterized by a large effective mass and temperature-dependent magnetic susceptibility. This feature has not been previously observed in U-5f Kondo lattice.

In this work, we scrutinized the alteration from disordered to ordered f-d Kondo hybridization in the paramagnetic phase of Kondo lattices of UTe2, USbTe and USbSe. We discovered that USbSe is in the coherent regime, exhibiting Pauli-like magnetic susceptibility and dispersive bands arising from the hybridization of U-6d and renormalized U-5f electrons. This leads to delocalized U-5f electrons, i.e., Bloch-like quasiparticle states, and a significant U-5f magnetic moment. We ascribed this duality of Pauli-like magnetic susceptibility and local magnetic moments to momentum reliant coherent f-d hybridization, with the VHS being the origin of the magnetic moment. This feature was employed to investigate the emergence of long-range magnetic ordering in UTe2 under pressure.

Results and discussion

Coulomb interaction tensor

The crystal structures of the orthorhombic phase (Immm) UTe230 and the tetragonal phase (P4/nmm) USbTe and USbSe31 are depicted in Fig. 1a. To investigate the impact of pressure on the electronic structure of UTe2, structures with a decreased volume of UTe2 were generated by reducing the experimental lattice parameters30 by 2 % (UTe2_2) and 4 % (UTe2_4). The constrained random phase approximation32 was employed to calculate the onsite Coulomb interaction UC and exchange interaction JH, for U-5f and U-6d orbitals, taking into account SOC. Figure 1b, c shows that both UC and JH increase and reach their unscreened values at high frequencies. The static UC of U-6d and U-5f orbitals were found to be comparable across the compounds studied. However, UC of U-5f is larger than that of U-6d, while JH for U-5f is smaller than that of U-6d. Supplementary Table S1 lists the static Slater’s integrals for all compounds. Ogasawara et al.33 reported JH of 0.58 eV for U-5f. This JH was observed in various uranium-based compounds, such as UPd3 and URu2Si2, through X-ray spectroscopy34,35. For UTe2, a JH value of approximately 0.51 eV was used to fit the results from resonant inelastic x-ray scattering (RIXS)36. However, our first-principles calculation for UTe2 yielded a significantly lower static JH of 0.24 eV. We propose that the on-site Coulomb and exchange interactions of the U-5f are material-specific, influenced by differences in atomic valence and screening effects. To illustrate, Ogasawara et al.33 calculated Slater’s integrals while considering the trivalent U3+ ion, whereas we did not assume a particular valence. Our ab initio calculation and experiment2 suggest that the 5f2 electronic configuration of U4+ is more probable in UTe2, as shown in Supplementary Fig. S3. These different atomic configurations should result in different values for JH. The potential causes for the discrepancy are discussed in Supplementary Table S1 and Fig. S2.

Fig. 1: Structure and calculated onsite Coulomb interaction.
figure 1

a Crystal structure of UTe2, USbTe, and USbSe. Calculated onsite (b) Coulomb interaction UC and (c) exchange interaction JH for U-5f (dashed lines) and U-6d (solid lines) with inclusion of spin-orbit coupling (SOC) as a function of νn = 2nπ/β bosonic frequencies. In (b), the inset shows magnified views of UC in the low frequency range.

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Electronic structures

The local self-energies of U-5f and U-6d were determined by solving two distinct single impurity models using the continuous time quantum Monte Carlo method within the framework of DMFT37. The dynamic Coulomb interaction tensors of U-5f and U-6d were employed in this process. It has been suggested that the frequency-dependent UC(ω) can lead to a narrower bandwidth38, resulting in a greater peak of f and more localization. The electron occupancies of U-5f and U-6d orbitals at 300 K are presented in Table 1. The U-6d orbitals were found to have a spread occupation across all d orbitals, while the U-5f orbitals were split into j = 5/2 and j = 7/2 multiplets, with a significant occupation of the j = 5/2 multiplet. The total occupations of U-5f orbitals were 2.27, 2.25, 2.26, 2.24, and 2.31 for UTe2, UTe2_2, UTe2_4, USbTe, and USbSe, respectively. The U-5f occupation for UTe2 was found to be in accordance with the measured 5f2 configuration39, indicating a substantial local magnetic moment of U-5f and the presence of an orbital selective Kondo effect40. For the f Kondo lattice, localization or itinerancy has been discussed in terms of several quantities, such as occupancy, density of states, and valence fluctuation41,42,43. Considering the total occupancy of f electrons in the 5/2 and 7/2 multiplets, we compared our results with those of δ-Pu in Supplementary Fig. S3.

Table 1 Electron occupancy
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Figure 2 presents the calculated spectral functions and atomic-orbital projected DOS. It is observed that all compounds share a common property of a strong U-5f peak in the vicinity of the Fermi level at low temperature. The temperature dependence of the U-5f peak between 150 K and 1000 K is consistent with that of the U-6d peak, indicating hybridization between U-5f and U-6d. In contrast, the DOS of other anions of Te, Sb, and Se are small and their temperature dependence are subtle. Furthermore, the presence of flat f bands and kink-like band structures implies that f-d Kondo hybridization governs the electronic structure in the vicinity of the Fermi level16,40,44. This is in agreement with the experimental observation of Kondo lattice in UTe245, USbTe5,46, and USbSe46. The calculated Kondo hybridization of USbTe with respect to temperature was found to be in agreement with angle-resolved photoemission spectroscopy (ARPES) measurements5. Similarly, the dispersive Kondo resonance peaks near the Fermi level of UTe2 were also observed in the ARPES47.

Fig. 2: Electronic structures.
figure 2

Calculated spectral functions and density of states (DOS) for (a) UTe2, (b) UTe2_2, (c) UTe2_4, (d) USbTe, (e) USbSe. f Orbital projected spectral functions for USbSe. f3 (d2) projected spectral function is presented in the upper (lower) panel (see the orbital labeling in Table 1). In the top of each figure, A denotes spectral weight. In each diagram of the DOS, solid lines are for a temperature of 150 K and dotted lines are for a temperature of 1000 K.

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Crossover from incoherent to coherent f-d Kondo cloud

The local total angular momentum susceptibility, \({\chi }_{loc}^{{J}_{Z}}\), was calculated as \(\int_{0}^{\beta }d\tau \langle {J}_{z}(\tau ){J}_{z}(0)\rangle\). Figure 3a highlights the \({\chi }_{loc}^{{J}_{Z}}\) of U-5f for all compounds. Whereas UTe2 and USbTe exhibits a Curie-like behavior, USbSe shows Pauli-like behavior. At temperatures between 150 and 250 K, a flat \({\chi }_{{J}_{Z}}\) was observed for USbSe which clearly shows in Fig. 3b. These results suggest that the magnetic moment on UTe2 is strong and resilient, while in USbSe, the electrons from the s, p, d and f orbitals of neighboring atoms are arranged in an antiparallel manner to the moment of U-5f, resulting in the cancellation of net moments48.

Fig. 3: Magnetic properties.
figure 3

a Calculated local total angular momentum susceptibility. Whereas UTe2 and USbTe exhibits a Curie-like behavior with incoherent f-d Hyb. (Kondo hybridization), USbSe shows Pauli-like behavior with coherent f-d Hyb. (Kondo Hybridization). b Inverse of the calculated local total angular momentum susceptibility. c Local angular moment correlation functions in the imaginary time τ. Calculated instantaneous J are presented with corresponding colors. d Temperature dependent frozen magnetic moments. The bar shows the ferromagnetic transition temperature TC = 127 K of USbTe and USbSe7.

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USbSe exhibits Pauli-type behavior from ~150 to 200 K, followed by Curie-like behavior at higher temperatures. This implies that USbSe is in the coherent regime with Pauli-type behavior at low temperatures. Our results of USbSe are reminiscent of δ-Pu48. The susceptibility of δ-Pu at its equilibrium volume is relatively flat, exhibiting Pauli-like behavior, below 600 K and then gradually decreases at temperatures higher than that. This Pauli-like behavior corresponds to the experimental measurements and is attributed to the coherence of the system. USbSe displays similar susceptibility to δ-Pu at its equilibrium volume, but with a transition temperature of about 250 K.

The local angular moment correlation functions \({\chi }_{{J}_{Z}}(\tau )=\langle {J}_{z}(\tau ){J}_{z}(0)\rangle\) are presented in Fig. 3c. The correlation function can be used to assess the degree of magnetic moment localization49. The instantaneous J values of U4+ range from 4.4 to 4.1, which is consistent with the 3H4 Russell Saunders ground state configuration50. Supplementary Fig. S4 shows that the charge susceptibility \({\chi }_{{{{{{{{\rm{N}}}}}}}}}=\int_{0}^{\beta }d\tau \langle N(\tau )N(0)\rangle\) of U-5f electrons is enhanced in the coherent regime, while the \({\chi }_{{J}_{Z}}(\tau )\) is reduced, as illustrated in Fig. 3c. These indicate that the itinerant character of the U-5f electrons in USbSe is significantly enhanced, as further evidenced by the diverging hybridization functions, as shown in Supplementary Fig. S5. USbSe exhibits a saturated value of ~2.0 for \({\chi }_{{J}_{Z}}(\tau )\) at τ → β/2, indicating that the U-5f electrons are localized (not completely screened) and thus a frozen magnetic moment is present. Figure 3d shows that the frozen magnetic moment of USbSe does not vanish entirely before the ferromagnetic transition upon cooling. Our research reveals that USbSe is in the coherent regime. USbSe exhibits Pauli-type behavior and a local magnetic moment at low temperature, indicating a dual character of the U-5f electrons, with both localization and itinerancy present.

Duality driven by momentum dependent f-d hybridization

We sought to identify the origin of the duality by calculating orbital-projected values. The six partially occupied U-5f orbitals can be divided into three distinct groups, fα = {f1, f6}, fβ = {f2, f5} and fγ = {f3, f4}, based on their self-energy and hybridization functions, as illustrated in Supplementary Fig. S5. Figure 4a displays the DOS, self-energies, and hybridization functions for f1 and f3, which correspond to fα and fγ, respectively. Figure 4b shows the quasiparticle weight, which is determined by the slope of the imaginary part of the self-energy, \(Z=1/(1-\frac{\partial Im\Sigma (i\omega )}{\partial i\omega }{| }_{i\omega \to {0}^{+}})\). The mass enhancement due to electronic correlations can be calculated as the inverse of the quasiparticle weight, \(\frac{{m}^{\star }}{m}=\frac{1}{Z}\)48,51. This Z is of interest, as it ranges from 1 (for weak electron correlation) to 0 (for strong electron correlation). Figure 4c shows α from the fitting of Im\(\Sigma ({{{{{{{\rm{i}}}}}}}}{\omega }_{n})\simeq -\Gamma +A{({{{{{{{\rm{i}}}}}}}}{\omega }_{n})}^{\alpha }\). Deviation from the linear variation, i.e., non-Fermi liquid behavior, can be attributed to the freezing of localized spin moments52. The DOS, self-energy, hybridization function, Z and α for partially occupied six f orbitals of all compounds were presented in Supplementary Figs. S5–S7.

Fig. 4: Orbital projected electronic properties.
figure 4

a DOS, imaginary part of self-energies (Σ) and hybridization functions (Δ) for f1 and f3 (see the orbital labeling in Table 1). Temperature dependence of (b) the quasi-particle weight Z factors and (c) calculated α from the fitting of Im\(\Sigma ({{{{{{{\rm{i}}}}}}}}{\omega }_{n})\simeq -\Gamma +A{({{{{{{{\rm{i}}}}}}}}{\omega }_{n})}^{\alpha }\), for f1 and f3. In (a), the inset shows separated views of the DOS in the vicinity of the Fermi level.

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Figure 4 illustrates that UTe2 has a higher Z factor than other compounds at elevated temperatures, implying a feeble electron correlation and consequently a small quasiparticle peak near the Fermi level. As the temperature is decreased, the Z factor decreases due to the emergence of incoherent f-d Kondo hybridization and a large f DOS in the vicinity of the Fermi level40, as shown in Fig. 4a, b. UTe2 also has a smaller α value compared to other materials, which is attributed to its more pronounced frozen magnetic moments.

Figure 4 shows that UTe2_4 and USbSe, which are in the coherent regime, have larger α values than UTe2, indicating smaller magnetic moments. At temperatures below 300 K, the Z factors of UTe2_4 and USbSe are larger than that of UTe2, indicating a suppression of electron correlations and an increase in the coherence energy of the electrons. This suggests a transition from an incoherent to a coherent behavior48. The transition is characterized by coherent f-d hybridization, resulting in an increased quasiparticle peak near the Fermi level. In contrast to UTe2, UTe2_4 and USbSe exhibit orbital selective hybridization. For UTe2, the hybridization function of f1 is significantly larger than that of f3. For UsbSe, the hybridization function of f3 at the Fermi level exhibits a strong divergence in comparison to f1. This results in a two peak structure of the DOS in the vicinity of the Fermi level, unlike the DOS of f1, as shown in Fig. 4a and Supplementary Fig. S8. This suggests that the degree of itinerancy is dependent on the orbital configuration. However, as shown in Fig. 4c and Supplementary Fig. S7, the orbital selective f-d hybridization does not have a significant effect on α for the six partially occupied f orbitals of USbSe. This indicates that the six f orbitals contribute similarly to the magnetic moment.

Figure 2e shows the DOS of USbSe, which exhibits a kink in the vicinity of the Fermi level, indicating the presence of a VHS. This is due to the flat bands in the six partially occupied f projected spectral functions, as seen in Fig. 2f and Supplementary Fig. S9. The flat f bands are subject to coherent f-d hybridization, which is momentum dependent and forms partially flat bands along X-M-Γ, Γ-Z-R and R-A-Z symmetry lines, resulting in the emergence of VHS. Figure 2f demonstrates this behavior. Near Γ and the R points, f3 and d2 form dispersing bands together, but at the Z point, the d2 bands do not exist and a flat f3 band is present, thus a VHS is created at the Z point. As can be seen in Supplementary Fig. S10, for USbSe, VHS appear close to the Fermi level at both A and M points on both DFT and DMFT bands. In DMFT, they are closer to the Fermi level due to electron correlation53. The momentum dependent f-d hybridization in uranium-based Kondo lattices gives rise to a dual character of Pauli-like magnetic susceptibility and local magnetic moments of U-5f electrons in the coherent regime. This duality is reminiscent of the behavior of d electrons in Ni metallic systems, which act both as localized moments and itinerant contributions due to the VHS28. Our results show that all partially filled U-5f orbitals contribute to local magnetic moments, which can be distinguished from OSMP or orbital-dependent hybridization to explain the dual nature. Table 1 and Fig. 2 demonstrate that the occupancy of U-5f orbitals in the J = 5/2 multiplet is away from half filling, indicating that the U-5f in the presented compounds is not described by Mott-physics. This is further evidenced by the metallic U-5f DOS shown in Fig. 2.

Investigating the magnetism of UTe2 under pressure

To gain a better understanding of the origin of long-range magnetic ordering of UTe2 at 1.5 GPa, we performed electronic structure calculations of UTe2 in its orthorhombic phase with reduced volumes. It was demonstrated in the preceding section that UTe2 with reduced volumes are in a coherent state. The temperature dependent \({\chi }_{loc}^{{J}_{Z}}\) shown in Fig. 3a further supports the transition. The overall decrease in \({\chi }_{loc}^{{J}_{Z}}\) with decreasing volume is in agreement with the experimental observation of a decrease in χa under pressure9. Similar to USbSe, this coherent f-d hybridization transition does not affect all f bands in the reciprocal space, resulting in partially flat f bands depending on momentum. This results in the emergence of VHS from the flat U-5f bands along the L-T-W and R-X1-Z symmetry lines, as shown in Fig. 2c. Our findings suggest that UTe2 is in a coherent state under pressure, with a VHS causing long-range magnetic ordering at low temperatures.

We discuss other potential causes of long-range magnetic ordering in two distinct scenarios of UTe2: fully-screened and under-screened. In the fully-screened case, the net local magnetic moment is zero. To induce long-range magnetic ordering, the local magnetic moment must appear. Figure 3c shows that the magnetic moment of U-5f decreases with decreasing volume, which is in agreement with the experimental observation of a decrease in effective moments under pressure reported in ref. 9. Consequently, the reduced magnetic moment of U-5f will remain fully-screened under pressure, with no impurities’ magnetic moment to be coupled.

In the case of under-screened UTe2, non-zero U-5f magnetic moments can interact through either exchange or RKKY interaction. It is reasonable to assume negligible direct exchange coupling between neighboring U-5f moments, as the f-f hybridization is negligible due to the small overlap of wave functions between U-5f. This is supported by our findings that f-d hybridization is dominant at low energy. We also suggest that RKKY interaction can be suppressed under pressure. Doniach’s work23 suggests that the transition from an antiferromagnetic to a Kondo-like state is driven by the competition between the binding energy of a Kondo singlet, \({W}_{K} \sim N{(0)}^{-1}{e}^{-1/N(0){J}_{0}}\), and that of the RKKY antiferromagnetic state, \({W}_{AF} \sim {J}_{0}^{2}N(0)\), where J0 is exchange coupling constant between the localized moments and the conduction electrons and N(0) is the density of conduction electron states. When J0 is below a certain threshold, the RKKY state is the most prominent, while above this, the Kondo singlet binding is the most influential. As shown in Supplementary Table S3, as the volume of UTe2 decreases, the spread of the Wannier functions of the d orbitals decreases, resulting in an increase in the bare UC and JH for U-6d as shown in Fig. 1b, c54. These results suggest that the RKKY interaction is weakened by increased J0 under pressure on UTe2. Consequently, the RKKY interaction is not a viable explanation for the emergence of long-range magnetic ordering in UTe2 under pressure.

This result cannot account for the lack of long-range magnetic order in UTe2 at ambient conditions. One possible explanation is that UTe2 remains in a fully-screened state until it transitions to the superconducting phase, not allowing RKKY interaction. Further investigation is required to evaluate this hypothesis and determine any magnetic instabilities.

Conclusions

When the chemical composition is altered or the lattice parameters are reduced, the uranium-based Kondo lattices undergo coherent f-d hybridization. This results in the emergence of Bloch-like quasiparticles from the renormalized U-5f. The duality of Pauli-like susceptibility and local magnetic moments is attributed to the van Hove singularity of the Bloch-like quasiparticles. This perspective on the magnetic properties of the Kondo lattice provides a theoretical basis to explain the mechanism of long-range magnetic ordering in Kondo lattices.

Methods

We employed ab initio linearized quasi-particle self-consistent GW (LQSGW) method combined with dynamical mean field theory (DMFT)37,55,56. This LQSGW+DMFT is based on simplified full GW+DMFT approach57,58,59,60. The electronic structure was calculated using LQSGW approaches61,62, and the local part of the GW self-energy was diagrammatically corrected within DMFT63,64,65.We also explicitly calculated the double-counting energy and Coulomb interaction tensor. Local self-energies for U-5f and U-6d were obtained by solving two different single impurity models, with spin-orbital coupling included in all calculations. The self energy is calculated on an imaginary frequency axis and is then analytically continued using a maximum entropy method66. Due to the computational challenges, we only consider the diagonal part of the self-energy, which is assumed to be diagonal in the spin-orbital coupled spherical harmonics basis. The crystal field (CF) effect is fully considered in the non-local LQSGW Hamiltonian. Based on this Hamiltonian, we build a local Hamiltonian in the spin-orbit coupled spherical harmonics basis and solve for the diagonal part of the self-energy. This approximation ignores the contribution of the off-diagonal CF effect to the local Hamiltonian. However, as shown in Supplementary Fig. S1, the off-diagonal components of hybridization are subtle, indicating that the off-diagonal CF effects on the local Hamiltonian can be neglected. For further details, approximation, and potential consequences, please see Supplementary Method 1 and Supplementary Note 1.