Direct bulk-sensitive probe of 5f symmetry in URu₂Si₂

Significance

The hidden order problem in URu₂Si₂ is an unanswered question in the field of strongly correlated electron materials. Although it has been studied for several decades, there is still no consensus about how this new phase forms. Understanding the hidden order phase formation is not only an intellectual problem, it will also advance concepts for designing quantum materials with new exotic properties. Many hidden order scenarios are based on the assumption of certain ground-state symmetries and the present study addresses this aspect. A spectroscopic technique, nonresonant inelastic X-ray scattering, that has become available through the use of high-brilliance synchrotrons, allows us to measure directly in a bulk-sensitive experiment the symmetry of the 5$f$ ground-state wave function in URu₂Si₂.

Abstract

The second-order phase transition into a hidden order phase in URu₂Si₂ goes along with an order parameter that is still a mystery, despite 30 years of research. However, it is understood that the symmetry of the order parameter must be related to the symmetry of the low-lying local electronic $f$-states. Here, we present results of a spectroscopic technique, namely core-level nonresonant inelastic X-ray scattering (NIXS). This method allows for the measurement of local high-multipole excitations and is bulk-sensitive. The observed anisotropy of the scattering function unambiguously shows that the 5$f$ ground-state wave function is composed mainly of the ${\mathrm{\Gamma }}_1$ with majority $J_z$ = $|4⟩$ + $|-4⟩$ and/or ${\mathrm{\Gamma }}_2$ singlet states. The incomplete dichroism indicates the possibility that quantum states of other irreducible representation are mixed into the ground state.

In heavy fermion rare earth or actinide compounds, the $f$ electrons are well localized at high temperatures, but as temperature is lowered, hybridization with conduction electrons becomes increasingly effective, resulting in a more itinerant $f$-electron character at low temperatures. These hybridized $f$ electrons form narrow bands and have large effective masses. Quasiparticle interaction effects in these narrow bands are responsible for the many exciting phenomena present in heavy fermion compounds: multipolar order (1), unconventional superconductivity (2), or quantum criticality (3). The hidden order phase in URu₂Si₂ is one example of the exotic low-temperature phases found in this material class. URu₂Si₂ is a tetragonal heavy fermion compound that undergoes two phase transitions, the nonmagnetic hidden order ($HO$) transition at $T_{HO}$ = 17.5 K, which involves an appreciable loss of entropy, and a superconducting transition at about 1.5 K (4–7). Below the $HO$ transition, small-ordered magnetic moments were observed in the earlier studies but were determined later to belong to a parasitic minority phase. With applied pressure (p $\ge$ 0.7 GPa), the $HO$ order is replaced by an antiferromagnetic phase with large-ordered moments (so-called $LMAF$-phase) (8). The order parameter of the $HO$ phase has been the subject of intense investigations for more than 30 years but, so far, has remained hidden, which has been the inspiration for its name. This second-order transition into an electronically ordered state involves a reconstruction of the Fermi surface (9, 10) and a change of quasiparticle scattering rate (11). The Fermi surfaces of the $HO$ and high-pressure $LMAF$ phase are very similar (12).

In URu₂Si₂ three-energy scales have been identified: a hybridization gap of ${\mathrm{\Delta }}_{hyb}\approx \mathrm{\hspace{0.17em}13}$ meV [150 K] (13) that opens below 27 K, another gap that opens in the $HO$ phase with ${\mathrm{\Delta }}_{HO}$ $\approx$ 4.1 meV [50 K] in the charge (9, 10, 14, 15) as well as spin channel (16, 17), and a resonance mode that appears in the $HO$ gap at $\cong$1.6 meV [18 K], also in both channels (18–20). Furthermore, with entering the $HO$ phase, the breaking of the fourfold rotational symmetry has been reported from torque experiments (21), high resolution X-ray diffraction on high quality crystals (22), and elastoresistance measurements (23). For a more detailed experimental and theoretical survey of physical properties of URu₂Si₂, we refer to the review article by Mydosh and Oppeneer (24).

In intermetallic actinide compounds, the valence state is often intermediate, and indeed, $N$-edge sum rules (25), lifetime reduced L-edge absorption (26), and soft X-ray photoelectron spectroscopy measurements (27) find a valence between 3⁺ and 4⁺ for URu₂Si₂. URu₂Si₂ is an itinerant system, and yet electron correlations on the U atom will reduce the charge fluctuations and favor also a particular local irreducible representation (28). In this respect, it is suggestive to assume that the U⁴⁺ $f^2$ configuration will give the dominant contribution, which is in line with first-principle dynamical mean-field theory (DMFT) calculations (29). The question is now which of the U⁴⁺ ($f^2$) states build up the itinerant state and lead to the formation of the $HO$. The present work presents the asymmetry of the inelastic X-ray scattering function S($\overrightarrow{q}$,$\omega$), as measured in a bulk-sensitive, nonresonant inelastic X-ray scattering (NIXS) experiment, and gives direct and quantitative information on the 5$f$ symmetry in URu₂Si₂.

To address the local 5$f$ degrees of freedom of URu₂Si₂, we will make use of the crystal-electric field (CEF) description of the U⁴⁺ $f^2$ configuration in $D_{4h}$ symmetry. The CEF splits the ninefold degenerate $J$ = 4 multiplet into five singlets and two doublets, which can be written in the following way when using the $J_z$ representation:

$${\mathrm{\Gamma }}_1^{(1)}(\theta )=\cos (\theta )|0⟩+\sin (\theta )\sqrt{\frac{1}{2}}(|4⟩+|-4⟩)$$

$${\mathrm{\Gamma }}_1^{(2)}(\theta )=\sin (\theta )|0⟩-\cos (\theta )\sqrt{\frac{1}{2}}(|4⟩+|-4⟩){\mathrm{\Gamma }}_2=\sqrt{\frac{1}{2}}(|4⟩-|-4⟩){\mathrm{\Gamma }}_3=\sqrt{\frac{1}{2}}(|2⟩+|-2⟩){\mathrm{\Gamma }}_4=\sqrt{\frac{1}{2}}(|2⟩-|-2⟩){\mathrm{\Gamma }}_5^{(1)}(\varphi )=\cos (\varphi )|\mp 1⟩+\sin (\varphi )|\pm 3⟩{\mathrm{\Gamma }}_5^{(2)}(\varphi )=\sin (\varphi )|\mp 1⟩-\cos (\varphi )|\pm 3⟩$$

Here, the values $\theta$ and $\varphi$ define the mixing of states that have equal irreducible representation, that is, the singlet states ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$ and doublet states ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$. The phase relations between the $J_z$ states are defined such that the operator $\widehat{J_x}$ is nonnegative. Note that ${\mathrm{\Gamma }}_1^{(1)}$(90°) = −${\mathrm{\Gamma }}_1^{(2)}$(0°) and ${\mathrm{\Gamma }}_5^{(2)}$(90°) = ${\mathrm{\Gamma }}_5^{(1)}$(0°) and, depending on the mixing angles $\varphi$ and $\theta$, the CEF states correspond to pure $J_z$ states (${\mathrm{\Gamma }}_1^{(1)}$(90°) $\iff |4⟩$+$|-4⟩$, ${\mathrm{\Gamma }}_2$ $\iff |4⟩$ − $|-4⟩$, ${\mathrm{\Gamma }}_1^{(2)}$(90°) $\iff |0⟩$, ${\mathrm{\Gamma }}_5^{(1)}$(90°) $\iff |\pm 3⟩$, and ${\mathrm{\Gamma }}_5^{(2)}$(90°) $\iff |\pm 1⟩$).

Determining CEF excitations and their symmetry in intermetallic U compounds is by no means trivial because the 5$f$ electrons are more itinerant than, for example, the 4$f$ electrons in the rare earth series, and the classic tool—inelastic neutron scattering—fails to observe sharp CEF excitations (30) due to dispersive effects and the large intrinsic widths that goes along with itinerant states. Nevertheless, there have been many experimental and also theoretical attempts to determine the symmetries of the 5$f$ ground state and low-lying electronic states in URu₂Si₂, and in literature, a wide spectrum of different scenarios can be found. The anisotropy of the static susceptibility is well described with a ${\mathrm{\Gamma }}_1^{(1)}$ singlet ground state, a ${\mathrm{\Gamma }}_2$ as a first excited state and the next states above 15 meV [170 K] (31). Analyses of elastic constant measurements find similar results (32). Also Kiss and Fazekas (33), Hanzawa (34), and Kusunose et al. (35) favor a ${\mathrm{\Gamma }}_1^{(1)}$, the model of Kiss and Fazekas being also compatible with a ${\mathrm{\Gamma }}_1^{(2)}$ singlet ground state (33), but these authors all propose different first excited states from their theoretical considerations. Haule and Kotliar (29) also propose two low-lying singlet states, a ${\mathrm{\Gamma }}_2$ singlet ground state and a ${\mathrm{\Gamma }}_1^{(2)}$ as a first excited state, a scenario that is compatible with the interpretation of polarized Raman studies that find a resonance at 1.6 meV in the $A_{2g}$ channel in the $HO$ phase (18, 19)*. Thermodynamic measurements by Santini and Amoretti (36) and resonant X-ray scattering data by Nagao and Igarashi (37) are interpreted in terms of a ${\mathrm{\Gamma }}_3$-singlet ground state with the ${\mathrm{\Gamma }}_1^{(1)}$ as a first excited state or alternatively with a ${\mathrm{\Gamma }}_5^{(1)}$ ground state (37). Another elastic constant study by Kuwahara et al. (38) yields a ${\mathrm{\Gamma }}_4$ as a lowest state. ${\mathrm{\Gamma }}_5^{(1)}$, and ${\mathrm{\Gamma }}_5^{(2)}$ doublets as ground states are concluded by thermodynamic studies of diluted URu₂Si₂ (39 and theoretical considerations by Ohkawa and Shimizu (40) and Chandra et al. (41). Finally $O$-edge X-ray absorption measurements by Wray et al. (42) favor the ${\mathrm{\Gamma }}_5^{(1)}$ and Sugiyama et al. the ${\mathrm{\Gamma }}_5^{(2)}$ doublet (43) as ground state.

There is clearly room for clarification. Hence, we aim at determining the symmetries of the ground state and low-lying states in URu₂Si₂ using a spectroscopic method that directly probes the U 5$f$ shell. We performed a core-level NIXS at the U $O_{\mathrm{4,5}}$-edges ($5d\to 5f$) with hard X-rays ($\approx$10 keV) and large momentum transfers ($\left|\mathbf{q}\right|$ $\approx$ 9.6 ${\mathrm{\AA }}^{-1}$). NIXS is a photon-in-photon-out technique that was used in the recent past on single crystals for determining the wave functions of cerium based systems (44, 45). In NIXS, the direction dependence of the momentum transfer $\overrightarrow{q}$ is used in analogy to the linear polarization dependence in an X-ray absorption spectroscopy (XAS) experiment (e.g., ref. 46 and also ref. 42) and accordingly “multipole selection rules” give access to the ground state symmetry (dipole for XAS). The higher multipoles that contribute significantly to the scattering function S($\overrightarrow{q}$,$\omega$) at large momentum transfers contain more information than dipole so that, e.g., asymmetries with higher than twofold rotational symmetry can be detected (44, 47). In addition, at the U $O_{\mathrm{4,5}}$-edge these excitations are significantly narrower than the dipole signal, which is strongly broadened due to the proximity of continuum states (48). Most importantly, it should be mentioned that a NIXS experiment does not involve an intermediate state so that the quantitative modeling is as straightforward as for XAS and the use of hard X-rays makes the signal truly bulk-sensitive in contrast to a soft XAS or soft resonant inelastic X-ray scattering (RIXS) experiment.

Results

Several NIXS studies, also on uranium compounds, show experimentally and theoretically how the multiplet excitations develop with increasing momentum transfer (44, 48–51). However, for convenience of the reader, we recapitulate briefly the principle of NIXS: when working at large-momentum transfers, the expansion of the transition operator exp(i$\overrightarrow{q}\overrightarrow{r}$) in spherical harmonics cannot be truncated after the first term, thus giving rise to excitations due to quadru-, octupole, and higher-order contribution in S($\overrightarrow{q}$,$\omega$). Fig. 1 shows the three nonvanishing contributions to S($\overrightarrow{q}$,$\omega$) calculated for the U $O_{4,5}$-edge; the radial part as function of momentum transfer in Fig. 1A and the isotropic spectra in Fig. 1B, each for the dipole and higher-multipole contributions. The excitations due to scattering from higher multipoles contribute substantially to the total intensity already for momentum transfers of $\left|\mathbf{q}\right|$ $\approx$ 9.6 ${\mathrm{\AA }}^{-1}$. Performing such an experiment on a single crystal and measuring the q-direction dependence will give S($\overrightarrow{q}$,$\omega$). This measurement then can be used for the CEF analysis where each state will have a specific direction dependence.

Fig. 1.

Momentum $|\overrightarrow{q}|$ dependence (A) and energy dependence (B) of the scattering function S($\overrightarrow{q}$,$\omega$) at the U $O_{\mathrm{4,5}}$-edge (5$d^{10}$4$f^2$ $\to$ 5$d^9$4$f^3$) for dipole (k = 1), octupole (k = 3), and dotriacontapole (k = 5) scattering orders. The gray vertical line marks the $\left|q\right|$-range of the experiment. Note: features above $\approx$106 eV appear unrealistically narrow because the proximity of continuum states is not accounted for.

Fig. 2A shows the simulation of S($\overrightarrow{q}$,$\omega$) of the $5d\to 5f$ transition ($O_{\mathrm{4,5}}$-edge) for the nine states of the $J$ = 4 ground state multiplet, i.e., the $f^2$ configuration. Contributions from other valence configurations are neglected. For more detailed information about the simulation, we refer to Materials and Methods. The spectra corresponding to the respective CEF states are calculated for the two directions $\widehat{q}\parallel$[100] and $\widehat{q}\parallel$[001], and some of the states exhibit a strong direction dependence. Here, $\theta$ and $\varphi$ are chosen such that the anisotropies are maximum, i.e., for the extreme cases of pure $J_z$ states (see definition of CEF states). The insets in Fig. 2A show the respective two electron 5$f$ charge densities. The charge densities of the pure states in Fig. 2A that appear rotational invariant do show lobes for $\theta$ and $\varphi$ $\ne$ 0 or 90° (Figs. S1 and S2).

Fig. 2.

(A–C) NIXS measurements of the U $O_{\mathrm{4,5}}$-edge for $\left|\mathbf{q}\right|$ = 9.6 ${\mathrm{\AA }}^{-1}$ and corresponding calculations for 5$d^{10}$4$f^2$ $\to$ 5$d^9$4$f^3$. (A) Simulation of S($\overrightarrow{q}$,$\omega$) of U crystal-field states for $J=\mathrm{\hspace{0.17em}4}$ in $D_{4h}$ symmetry for the two directions $\widehat{q}\parallel$[100] (blue) and [001] (red). Insets show the corresponding electron densities (Materials and Methods). (B) NIXS data for momentum transfers $\widehat{q}\parallel$[100] (blue) and [001] (red) at T = 25 K. (C) Dichroism at 25 K in percentage defined as difference ${\text{I}}_{\widehat{q}\parallel [100]}$ − ${\text{I}}_{\widehat{q}\parallel [001]}$ relative to peak height R₁ as defined in the isotropic data (Fig. 3), data (black dots) and calculations (green lines) for the crystal-field states with the correct sign of dichroism. Here the data points have been convoluted with a Gaussian of 0.5-eV FWHM.

Fig. S1.

NIXS simulations for momenta $\widehat{q}$ parallel to [100] (blue) and [001] (red) for the two states ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$($\theta$) (A) and ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$($\varphi$) (B) for different $J_z$ admixtures expressed in terms $\theta$ and $\varphi$.

Fig. S2.

NIXS simulations of U $O_{\mathrm{4,5}}$ for momenta $\widehat{q}$ parallel to [100] and for $\widehat{q}$ turned toward [010] by 22.5° and 45°, i.e., $\widehat{q}\parallel$[100] (blue), $\widehat{q}\parallel$[520] (dark green), and $\widehat{q}\parallel$[110] (light green) for the two states ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$($\theta$) (A) and ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$($\varphi$) (B) for different $J_z$ admixtures expressed in terms of $\theta$ and $\varphi$.

The NIXS experiment was performed at large momentum transfers (Materials and Methods, Experimental Setup) so that the signal is dominated by higher-order scattering (beyond dipole). Data were taken below and above the $HO$ transition at 5 K and 25 K and with successively rising temperature up to 300 K. All data shown are background corrected.

Before discussing the direction dependence, we first show in Fig. 3 the isotropic data (see Materials and Methods for construction of isotropic spectra), together with a simulation using the ionic 5$f^2$ configuration for the U. We can clearly observe a very good agreement, thereby establishing that the spectrum is dominated by the atomic multiplet structure. This observation is important because it validates a posteriori the use of local probes (such as core level NIXS) to test models using also local irreducible representations. Perhaps more surprising is that a single ionic configuration can reproduce the spectrum so well despite the known covalency of the U. However, it has been explained by, for example, Gunnarson and Schönhammer (52) as well as de Groot (53), that $d\to f$ and $p\to d$ core-level XAS (NIXS) for 4$f$/5$f$ and 3$d$ compounds, respectively, highlights the spectral weight of the energetically lowest-lying (and major) configuration at the expense of those of the higher-lying configurations, making the technique extremely powerful for determining the symmetry of the ground state (Materials and Methods, Spectroscopy).

Fig. 3.

Experimental (black dots) and simulated (red line) isotropic spectrum of URu₂Si₂ at the U $O_{\mathrm{4,5}}$-edge (5$d^{10}$4$f^2$ $\to$ 5$d^9$4$f^3$) for T $\le$ 25 K. For details, see Materials and Methods, Isotropic Spectrum.

We now discuss the direction dependence of the data measured above the $HO$ transition because they are not affected by any possible impact of the $HO$. In Fig. 2B, the NIXS data of URu₂Si₂ at 25 K are shown for the in-plane direction $\widehat{q}\parallel$[100] (blue dots) and out-of-plane direction $\widehat{q}\parallel$[001] (red dots). The error bars reflect the statistical error. There is a large anisotropy that can be directly compared with our simulations.

A more detailed comparison of data and simulations excludes immediately the ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$($\theta$) states with strong $J_z=|0⟩$ contributions, the ${\mathrm{\Gamma }}_3$ and ${\mathrm{\Gamma }}_4$ singlets with $J_z=|2⟩$ and $J_z=|-2⟩$, as well as the ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$($\varphi$) states with strong $J_z=|\pm 1⟩$ weight. Actually, only singlet states with majority $|+4⟩$ and $|-4⟩$ or a doublet with majority $|\pm 3⟩$ show the correct direction dependence, i.e., red over blue (Fig. 2A). To be more quantitative, we compare the measured dichroism of about 35% (Fig. 2C) with the simulated dichroism of the URu₂Si₂ wave functions in question. Here, the dichroism is defined as the difference of the intensities for $\widehat{q}\parallel$[100] and $\widehat{q}\parallel$[001] relative to the peak height R₁ of the isotropic spectrum (Fig. 3). We find that ${\mathrm{\Gamma }}_1^{(1)}$(90°) [or ${\mathrm{\Gamma }}_1^{(2)}$(0°)] and also ${\mathrm{\Gamma }}_2$ reproduce the size of the anisotropy quite well, although their dichroism is with 43 or 50% slightly larger than the measured value. A ${\mathrm{\Gamma }}_1^{(1)}$ state of majority $J_z=|4⟩$ and $|-4⟩$ symmetry, but with some $J_z=|0⟩$ [${\mathrm{\Gamma }}_1^{(1)}$(70° or 110°)] would produce a slightly smaller dichroism of about 36% (Fig. 2C).

The ${\mathrm{\Gamma }}_5^{(1)}$(90°) [or ${\mathrm{\Gamma }}_5^{(2)}$(0°)], i.e., the doublet states with the highest amount of $J_z$ = $|\pm 3⟩$, do not yield sufficient dichroism: the dichroism of 9% is by a factor of 4 too small and would decrease further or even change sign with increasing amount of $|\pm 1⟩$. [In the following, we skip writing out the ${\mathrm{\Gamma }}_i^{(2)}$ alternative state because of ${\mathrm{\Gamma }}_i^{(1)}$(90°) = −${\mathrm{\Gamma }}_i^{(2)}$(0°), i = 1 or 5.]

Fig. 4A shows the $ac$ asymmetry of the scattering function for all temperatures. Also, here, the error bars reflect the statistical error. We find that within the error bars the 5 K and 25 K are identical. We further find that there is no change with temperature up to 300 K as is demonstrated by plotting the dichroism for all temperatures in Fig. 4B. The Boltzmann population with temperature of any state other than the ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°), and ${\mathrm{\Gamma }}_2$ state will change the direction dependence of the scattering (compare Fig. 2A). Hence, we conclude from the absence of any changes in the spectra up to 300 K that the ground state consists mainly of the ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°) or the ${\mathrm{\Gamma }}_2$ singlet or that one of the two singlets forms the ground state with the respective other state close in energy. We can further estimate from the impact of thermal occupation that the states with weak dichroism like the ${\mathrm{\Gamma }}_5^{(1)}$(90°), ${\mathrm{\Gamma }}_3$ and ${\mathrm{\Gamma }}_4$ must be higher than 150 K (13 meV), whereas states with stronger opposite anisotropy must be even higher in energy.

Fig. 4.

(A) Temperature dependence of the URu₂Si₂ U $O_{\mathrm{4,5}}$-edge NIXS spectra for $\widehat{q}\parallel$[100] (blue dots) and $\widehat{q}\parallel$[001] (red dots). For better comparison, the $T$ dependent data are underlain with the difference in spectral weight of the 5 K data. (B) Dichroism ${\text{I}}_{\widehat{q}\parallel [100]}$-${\text{I}}_{\widehat{q}\parallel [001]}$ for all temperatures, convoluted with a Gaussian of 0.5-eV FWHM. (C) NIXS data and simulations (5$d^{10}$4$f^2$ $\to$ 5$d^9$4$f^3$) for in-plane momenta $\widehat{q}$ parallel to [100] and for $\widehat{q}$ turned toward [010] by 22.5° and 45°, i.e., $\widehat{q}\parallel$[100] (blue), $\widehat{q}\parallel$[520] (dark green), and $\widehat{q}\parallel$[110] (light green) for all CEF states.

Fig. 4C shows data taken within the plane, for $\widehat{q}\parallel$[100] and for two directions 22.5° and 45° toward [010], as well as the respective simulations for all CEF states. Neither below nor above the $HO$ order transition can we resolve any anisotropy within the statistical error bar. This result does not contradict our previous findings that either one of the two singlet states ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°) and ${\mathrm{\Gamma }}_2$ forms the ground state because the asymmetries expected from simulations are rather small and most likely covered by statistics of this low count experiment. The in-plane data even confirm the out-of-plane data when comparing the measured and simulated shape of the spectra: for example, the peak at 94 eV is clearly smaller than the peak at 97 eV for the simulated ${\mathrm{\Gamma }}_1^{(1)}$(90°) and ${\mathrm{\Gamma }}_2$ spectra (gray lines in Fig. 4C), in agreement with the data, whereas the two peaks are about the same for all other states. For the in-plane simulation for different values of $\theta$ and $\varphi$, we refer to Fig. S2.

Discussion

Our results of a ground state that consists mainly of ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°) and/or ${\mathrm{\Gamma }}_2$ agree well with the description of the anisotropy of the static susceptibility (31) and the analysis of the temperature dependence of the elastic constants (32), which are well described with a ${\mathrm{\Gamma }}_1^{(1)}$ of majority $J_z$ = +4 and −4, a ${\mathrm{\Gamma }}_2$ as a first excited state, and another state above 150 K. Our results also confirm DMFT calculations that finds these two singlet states as low-lying states close in energy [ref. 29 and supplementary materials in Kung et al. (19)], but the experiment yields the additional information that the $J_z$ = +4 and −4 in the ${\mathrm{\Gamma }}_1$ is dominating. We further would like to stress that linear polarized XAS data at the U $O_{\mathrm{4,5}}$-edge (42) also agree with our findings in the sense that both, the NIXS and XAS dichroism, rule out the ${\mathrm{\Gamma }}_1^{(2)}$(90°) [or ${\mathrm{\Gamma }}_1^{(1)}$(0°)], the ${\mathrm{\Gamma }}_3$, or the ${\mathrm{\Gamma }}_4$ as possible ground states and find no temperature dependence across the $HO$ transition. The smaller direction dependence that lets the authors of ref. 42 assign the ${\mathrm{\Gamma }}_5^{(1)}$ doublet as ground state might be due to the higher surface sensitivity of the XAS experiment.

A pure ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°) or ${\mathrm{\Gamma }}_2$ or both close in energy does confront us with the dilemma that neither would break the $C_4$ in-plane symmetry, as suggested by the torque (21), high precision X-ray (22), and elastoresistance (23) results nor would an ordering out of a singlet state yield sufficient loss of entropy across the $HO$ transition. To allow for a rank-5 $E^{-}$ $HO$ parameter, as in the fully microscopic itinerant approach (54, 55), the inclusion of the twofold degenerate CEF state of $E$-type is a necessity in the present more localized picture. Also, the interpretation of the resonance intensity in the main A_2g and other Raman channels in terms of a staggered chirality density wave requires a mixing of ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ singlet states that support a hexadecapole-type hidden order parameter. The model of Kung et al. (19) contains both dominant A_2g and subdominant B_1g symmetry parts. The latter involves higher energy CEF states and coupling to the lattice leads to a secondary orthorhombic distortion that leaves only twofold symmetry. As a result, resonance intensity leaks into forbidden channels.

Our experiment shows that the CEF components are mainly of the ${\mathrm{\Gamma }}_1^{(1)}$(90°) or ${\mathrm{\Gamma }}_2$ singlet type, but we did not observe the theoretically maximum possible dichroism (Fig. 2C), so that the data allow the presence of some other symmetry. The mixing of an irreducible representation, other than ${\mathrm{\Gamma }}_1^{(1)}$(70°–90°–110°) or ${\mathrm{\Gamma }}_2$, into the ground state cannot rely on Boltzmann occupation because that would have been observable in the temperature dependence of the NIXS data. However, a Kondo-type mechanism where an $f^3$ Kramers doublet hybridizes with the CEF manifold of the energetically more favorable $f^2{ϵ}_k^{+1}$ configuration is feasible for constructing a ground state with different $f^2$ CEF characters. Here, ${ϵ}_k^{+1}$ denotes an electron in the host conduction band. We recall that a hybridization gap of 13 meV opens up below 27 K (13), which is also seen in the DMFT calculation that includes these Kondo processes (19, 29). The stabilization energy of this Kondoesque wave function should be of the same order as the hybridization gap and the contributing CEF configurations should also be within this energy range.

Summary

The bulk-sensitive, U $O_{\mathrm{4,5}}$ NIXS data of URu₂Si₂ exhibit the atomic multiplet structure of the $f^2$ configuration. The huge out-of-plane anisotropy shows that the symmetry of the ground state consists mainly of the ${\mathrm{\Gamma }}_1^{(1)}$(90°) or ${\mathrm{\Gamma }}_2$ singlet states in the U⁴⁺ ($f^2$) configuration and/or that these two states are close in energy. The data do not exhibit any temperature dependence, neither across the $HO$ phase transition nor in the temperature interval up to 300 K, the latter setting constraints to the proximity of next-higher excited states. Scenarios for constructing a ground state that is a superposition of different irreducible representations without relying on Boltzmann statistics are discussed.

Materials and Methods

Samples.

A high-quality single crystal of URu₂Si₂ was grown with the traveling zone method in the two-mirror furnace in Amsterdam under high-purity (6N) argon atmosphere. The crystal was checked and oriented with X-ray Laue diffraction for its single-crystalline nature. The oriented crystal was cut using the spark-erosion method, after which the relevant surfaces [(100), (110), and (001)] were polished. A bar-shaped piece of the single crystal was characterized by resistance measurements.

Experimental Setup.

The scattering function S($\overrightarrow{q}$,$\omega$) was measured in a NIXS experiment at the beamline ID20 at European Synchrotron Radiation Facility (ESRF). Two monochromators [Si(111) and Si (311)] set the incident energy to 9,690 eV, and the scattered intensity was analyzed by one column of three Si(660) crystal analyzers at an in-plane scattering angle of 2$\vartheta$ = 153° and detected in a Maxipix 2D detector with an overall energy resolution of about 0.8 eV. This setting corresponds to a momentum transfer of $\left|\mathbf{q}\right|$ = 9.6 ${\mathrm{\AA }}^{-1}$. The crystals with (100), (110), and (001) surfaces allowed realizing $\widehat{q}\parallel$[100], [110], and [001] in specular geometry and also other directions when going off specular. It turned out that specular geometry (same path for photon in as for photon out) is not necessary because $\widehat{q}\parallel$[110] measured specular on the (110) crystal and 45° off specular on the (100) crystal gave the same result. For cooling, the samples were mounted in a He flow cryostat. The elastic line was measured before each setting to determine the zero energy transfer and exact instrumental resolution for each analyzer. The spectra of the U $O_{\mathrm{4,5}}$-edges were then normalized to their pre-edge intensity. Scans over a wide energy range were taken to correct for the Compton scattering and some minor constant background. The Compton background was fitted to a Gaussian and then subtracted from the data.

Spectroscopy.

Why does $O$-edge XAS or NIXS resemble the $f^2$ multiplet structure and why is $O$-edge XAS or NIXS sensitive to the symmetry? The energy order of the local configurations of the ground state and of the NIXS/XAS core-hole final state problem is identical, so that the spectral weights of the other local configurations are strongly suppressed due to quantum mechanical interference effects. These interrelations are well explained, for example, by Gunnarsson and Schönhammer (figure 7 in ref. 52) for the case of Ce $M_{4,5}$ XAS and by de Groot (53) for the case of 3$d$ transition metal $L_{2,3}$ XAS. A quote from the latter is as follows: “XPS is sensitive to the charge transfer effects…while XAS is sensitive to the symmetry of the ground state with its characteristic multiplet” (ref. 53, pp. 549 and 550). It is important that the relevant intra- and intershell Coulomb interactions are of similar size to have the same order of configuration energies in the ground state and core-hole final state problem: this is true for the Ce $M_{4,5}$ (3$d$, 4$f$), U $O_{4,5}$ (5$d$, 5$f$), and transition metal $L_{2,3}$ (2$p$,3$d$) edges but not for the Ce $L_{2,3}$ (2$p$,5$d$ with 4$f$ as spectator) or U $L_{2,3}$ (2$p$,6$d$ with 5$f$ as spectator) because the Coulomb interaction with the 5$d$ or 6$d$ is negligible.

Simulations.

The simulations include spin–orbit coupling and Coulomb interaction and are based on an ionic model with a U⁴⁺ 5$f^2$ configuration. The atomic values are calculated with the Cowan code (56), but the Slater integrals for Coulomb interactions are reduced by a constant factor to account for the screening of the moments in the solid. The 5$f$–5$f$ and 5$d$–5$f$ reduction was adjusted to about 50% to match the experimental energy spread of the multiplet signal of the isotropic data in Fig. S1 (for construction of isotropic spectrum, see below). The ratio of multipole contributions was slightly adjusted by varying $|\mathbf{q}|$ (49). In the simulations, the actual value for $|\mathbf{q}|$ was slightly larger than according to the experimental scattering triangle because the radial part of the wave functions that enter the calculations is based on the atomic values. For all finite values of spin–orbit coupling and Coulomb interaction the $J$ = 4 multiplet forms the Hund’s rule ground state. The relative contributions of different angular momenta $L=$3, 4, 5 depend on the ratio of spin–orbit coupling and Coulomb interaction and are 1, 14, and 85% for our reduction factors, respectively.

Within the $J=\mathrm{\hspace{0.17em}4}$ basis we, create the local eigenstates (${\mathrm{\Gamma }}_1$ to ${\mathrm{\Gamma }}_5$) restricted to the $f^2$ configuration by combining different states of $J_z$ considering the constraints by group theory. The Hamiltonian includes the local spin–orbit coupling and multipolar Coulomb interaction, which are much larger (up to 1 Rydberg) than the final-state core-hole lifetime (order of 1 eV) but neglects the effects of crystal-field, covalent bonding and band formation of the crystal, which will be smaller or of the same order of magnitude as the core-hole lifetime. The calculations are performed using the Quanty code (57). To account for instrumental resolution, lifetime effects, and interference effects with the continuum the multiplet lines are broadened with a Gaussian [full-width at half-maximum (FWHM)$=\mathrm{\hspace{0.17em}0.8}$ eV], a Lorentzian (FWHM $=\mathrm{\hspace{0.17em}1.3}$ eV), and a Mahan-type line shape (with an asymmetry factor 0.18 and an energy width of the continuum of 1,000 eV) to mimic the asymmetry due to the itinerancy.

Charge Densities.

The charge densities are calculated for two electrons. The surface thus tells how much charge can be found in a specific direction. A charge density for a one-electron system would be an occupied orbital and a “sum” of different orbitals when the system can be represented by a single Slater determinant. The color is related to the spin density with up as red, down as blue, and zero as gray (Fig. 2A). For URu₂Si₂, ${LSJJ}_z$ coupling is often assumed whereby $L$ $S$ $J$ and $J_z$ are all good quantum numbers. Doing so results in density plots with many more features than in the present report. This assumption is basically equivalent to saying that $F_2$, $F_4$, and $F_6$ Slater integrals are infinitely larger than the spin–orbit coupling. That approximation is not valid, and spin–orbit coupling mixes states $LS$ with states $L$+1 $S$-1 or $L$-1 $S$+1. This mixing is included in our calculations. Furthermore, we know that both the multipole part of the Coulomb interaction as well as the spin–orbit interaction are not (not really) screened in a solid. In other words, the U 5$f$ shell in URu₂Si₂ is between $LS$ and $jj$ coupling and was taken into account.

Isotropic Spectrum.

The isotropic spectrum is given by the trace of the conductivity tensor. For dipole transitions (k = 1), this tensor can be written as a 3 × 3 matrix with two independent diagonal elements in $D_{4h}$ symmetry. However, for higher multipoles, the conductivity tensor has also a higher dimension, i.e., 7 × 7 for octupole (k = 3) and 11 × 11 for dotriacontapole (k = 5). Here, we obtain the experimental isotropic spectrum containing these three relevant conductivity tensors by combining 10 independently measured directions. The calculated isotropic spectrum is obtained by averaging over all CEF states. The red line in Fig. 3 is the simulation of the isotropic data after optimizing the respective parameters.

SI Text

Out-of-Plane Anisotropy.

Fig. S1 A and B shows the out-of-plane anisotropy in the U $O_{\mathrm{4,5}}$-edge NIXS spectra of the ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$($\theta$) and ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$($\varphi$) wave functions for values of $\theta$ and $\varphi$ between 0 and 90°. Fig. S1, Insets show the respective charge densities. Please note that −${\mathrm{\Gamma }}_1^{(1)}$(90°) = ${\mathrm{\Gamma }}_1^{(2)}$(0°) and ${\mathrm{\Gamma }}_1^{(2)}$(90°) = ${\mathrm{\Gamma }}_1^{(1)}$(0°). The same holds for the ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$, i.e., −${\mathrm{\Gamma }}_5^{(1)}$(90°) = ${\mathrm{\Gamma }}_5^{(2)}$(0°) and ${\mathrm{\Gamma }}_5^{(2)}$(90°) = ${\mathrm{\Gamma }}_5^{(1)}$(0°).

In-Plane Anisotropy.

Fig. S2 A and B shows the in-plane anisotropy in the U $O_{\mathrm{4,5}}$-edge NIXS spectra of the ${\mathrm{\Gamma }}_1^{(\mathrm{1,2})}$($\theta$) and ${\mathrm{\Gamma }}_5^{(\mathrm{1,2})}$($\varphi$) wave functions for values of $\theta$ and $\varphi$ between 0 and 90°. Insets show the respective charge densities. According to simulations, none of the states exhibits a noteworthy in-plane anisotropy in the scattering intensity, although the charge densities appear anisotropic and there are multipole contributions in S($\overrightarrow{q}$,$\omega$). This finding can be understood when considering that the scattering process at the U_4,5-edge is from 5$d\to$ 5$f$, i.e., we scatter from a state with a complicated shaped charge distribution to another one with a nonsimple shape.