CeRu4Sn6: a strongly correlated material with nontrivial topology

Topological insulators form a novel state of matter that provides new opportunities to create unique quantum phenomena. While the materials used so far are based on semiconductors, recent theoretical studies predict that also strongly correlated systems can show non-trivial topological properties, thereby allowing even the emergence of surface phenomena that are not possible with topological band insulators. From a practical point of view, it is also expected that strong correlations will reduce the disturbing impact of defects or impurities, and at the same increase the Fermi velocities of the topological surface states. The challenge is now to discover such correlated materials. Here, using advanced x-ray spectroscopies in combination with band structure calculations, we infer that CeRu4Sn6 is a strongly correlated material with non-trivial topology.

tetragonal crystal-electric field (CEF) 26,28 . μSR measurements did not find magnetic order down to 50 mK, thus pointing out the importance of Kondo-type interactions in this material 30 .
One of the pressing open questions is whether CeRu 4 Sn 6 fulfills the conditions mentioned in the theoretical studies about strongly correlated topological insulators [5][6][7][8][9][10] . To answer this question, we determine here whether the hybridization between conduction and 4f electrons (cf-hybridization) in CeRu 4 Sn 6 is strong enough and whether the CEF ground state has a favorable symmetry to develop a gap of sizable magnitude. The degree of hybridization has been investigated by performing a temperature dependent x-ray absorption spectroscopy (XAS) experiment at the Ce L-edge to obtain the Ce valence and the Kondo temperature T K . The CEF ground state symmetry was studied by measuring the Ce M-edge in an XAS experiment and the Ce N-edge in a non-resonant inelastic x-ray scattering (NIXS) experiment. We have also performed band structure calculations to understand better our experimental findings and to test the robustness of our conclusions. Details about the experimental set-ups and the single crystal preparation and quality can be found in the Section Methods.

Results
Cerium 4f valence and Kondo temperature. Trivalent cerium (Ce 3+ ) has a 4f occupancy n f = 1 (f 1 ). The presence of strong cf-hybridization leads to a partial delocalization of the 4f electrons into the conduction electron sea and the resulting non-integer valent 4f ground state can be written in a local picture as a mixed state , whereas c 0 2 quantifies the degree of delocalization. The hybridization is described in terms of the formation of a Kondo singlet state at k B T K below the energy of the 4f 1 configuration, which breaks down as thermal fluctuations match the same energy. Hence, information about the Kondo temperature T K is obtained from the temperature variation of n f , or equivalently of c 0 2 if  c c 2 2 1 2 or if c 2 2 does not change with T 32-34 . Core level spectroscopy is a powerful tool for determining the occupations of the various 4f n configurations 31 . This is based on the fact that the impact of the core-hole potential depends strongly on the configuration. Moreover, recent XAS experiments measuring the α L 1 recombination process ( → ) 3 2 with partial fluorescence yield (PFY-XAS) at the Ce L 3 -edge ( → ) p d 2 5 have tremendously improved the accuracy of determining valencies [34][35][36] . We have performed such a PFY-XAS experiment on CeRu 4 Sn 6 , determined c 0 2 and obtained T K from its temperature dependence. Figure 1(a) shows the PFY-XAS data of CeRu 4 Sn 6 at 5 K (open circles). The strongest absorption is due to the f 1 contribution in the ground state, the small but well resolved hump at about 5740 eV is due to the f 0 contribution, and the small shoulder on the low energy tail of the f 1 peak (just above 5720 eV) is due to some f 2 spectral weight. It is not straightforward to describe the L 3 -edge spectral line shape since it is determined by the unoccupied 5d density of states in the presence of a 2p core hole. Yet, for extracting the f n weights, it is sufficient to fit the data with a set of Gaussian or similar functions as a mean to obtain their integrated intensities [37][38][39] . We have applied such a fit by describing each spectral weight (I(f 0 ), I(f 1 ), and I(f 2 )) with identical line shapes, consisting of three Gaussian profiles each. Only the total intensities and central energy positions were varied. Each profile includes a tanh-type step function (black lines) with the maximum slope at the peak maximum to fit the background. The background was adjusted so that the modulations due to the extended absorption fine structure (EXAFS) at ≈ 5749, ≈ 5754, and ≈ 5759 eV are averaged out. The red line in Fig. 1 under the assumption ( )  f c I n n 2 . This is an approximation since it neglects hybridization effects in the final states 39 . As in core level photoemission 40 , I(f 2 ) overestimates c 2 2 and I(f 0 ) underestimates c 0 2 so that the above values for n f and c 0 2 should be taken as upper and lower limits, respectively. It is therefore safe to conclude that the cf-hybridization is very strong in CeRu 4 Sn 6 . Figure 1(b) shows the temperature dependence of the PFY-XAS data, normalized to the integrated intensity. The inset in Fig. 1(b) shows the f 0 region on an enlarged scale after smoothing the data, and it is clearly seen that the main changes occur within the temperature interval of 150 to 250 K.  4 44 . We did not use inelastic neutron scattering since Kondo insulators exhibit spin gaps and strongly broadened magnetic excitations [45][46][47][48] , complicating the extraction of magnetic spectral weights which are required for the determination of wave functions. There are some neutron data on CeRu 4 Sn 6 which also exhibit a spin gap-type response that peaks at 30 meV and broadened magnetic excitations above it in the high energy data plus some quasielastic intensity in the low energy data 49 . In contrast, XAS and NIXS target the ground state symmetry directly: In the soft XAS experiment, the dipole selection rules for linearly polarized light give access to the initial state symmetry. Excited CEF states only contribute to the polarization as they get increasingly populated with rising temperature. A NIXS experiment with hard x-rays is truly bulk sensitive, and Scientific RepoRts | 5:17937 | DOI: 10.1038/srep17937 when performed at large momentum transfers  q the signal is dominated by higher multipole orders of the scattering cross section 50,51 . In the NIXS experiment the  q-dependence (  q-direction with respect to the sample orientation) replaces the polarization dependence in XAS and, accordingly,  q-dependent multipole selection rules give access to the ground state symmetry 44 . Figure 2 shows the soft XAS data at the Ce M 4,5 edge for the two polarizations E [001] and ⊥ E [001] for several temperatures between 4 and 300 K. At all temperatures there is a strong polarization effect which decreases only slightly between 200 and 300 K, thus indicating the first excited CEF state is situated at ≥ 25 meV. The data at 4 K will therefore yield ground state information. In addition to the main absorption line, there are smaller humps at slightly higher photon energies (see arrows) due to some f 0 weight in the initial state. Its spectral weight decreases with rising temperature in agreement with the findings from the PFY-XAS data. Figure 3(a) shows the linearly polarized low-temperature data (red and blue dots). The spectra are simulated ionically with a full muliplet code (see Methods, Analysis) to account for the f 1 ground state symmetry. Transitions due to f 0 in the ground state or transitions to continuum states are not accounted for. Figure 3(a) shows a simulation assuming a pure / , ± / = Γ 5 2 1 2 6 ground state (red and blue lines), which is the state with rotational symmetry [see inset in Fig. 3(a)]. The simulation reproduces the main features of the spectra and their polarization dependence, but the match is by no means perfect. This issue is investigated further in the following.
In Fig. 3 001 , black dots) is compared with the simulated LD of the three pure / , J 5 2 z states (colored lines). Note here the f 0 and continuum contributions have no or negligible polarization dependence. The LD of the pure / , ± / = Γ 5 2 1 2 6 (orange) and also of the / , ± / 5 2 3 2 (blue), corresponding to a Γ 7 -symmetry with α = 0, have the correct sign, but the LD of the latter is too small and any / , /  5 2 5 2 -admixture (green) in a Γ 7 -type ground state would make the discrepancy even larger. Even if we adopt a Γ 7 -type ground state of predominantly / , ± / 5 2 3 2 , the measured LD will be larger than the simulated one. We therefore conclude that although the ground state is not a pure Γ = / , ± / 5 2 1 2 6 , it is still mainly of that character.
A NIXS experiment at the Ce N-edge was performed in order to rule out surface effects being responsible for the unsatisfactory agreement between the soft XAS data and the simulation. The data in Fig. 4 (red and blue dots) were obtained after 240 s counting per point and after subtracting a rising, linear background (Compton). The   The strongest dichroism in the data appears at the shoulder structures at about 105 and 113 eV. Qualitatively the simulation for the / , ± / 5 2 1 2 state reproduces the data quite well. In contrast, the / , ± / 5 2 3 2 state suggests a crossing of both directions on both shoulders, which is not observed experimentally, and the dichroism of the / , ± / 5 2 5 2 state has the wrong sign. We therefore conclude also from this experiment that the ground state must be governed by a Γ = / , ± / 5 2 1 2 6 symmetry, but at the same time is not purely given by the Γ 6 .

Discussion
What could be the reason for the deviation between the experiment and the simulation starting from a pure Γ = / , ± / 5 2 1 2 6 ? Already in 1986 van der Laan et al. pointed out that in the presence of strong hybridization higher lying states contribute to the M-edge signal 52 . In CeRu 4 Sn 6 we find that the Kondo temperature (170 K) is not much smaller than the CEF excitations (≥ = )  25meV 290K so that a contribution of higher lying CEF states to the ground state is indeed to be expected. Given the fact that the LD of these higher lying CEF states is smaller or even of opposite sign, see Fig. 3(b,c), it is evident that the LD of this composite ground state will be reduced from that of the pure / , ± / 5 2 1 2 state. We could also reverse the argumentation: the observation of a reduced LD can be taken as evidence for the presence of strong hybridization. Nevertheless, we can conclude that the ground state of CeRu 4 Sn 6 is predominantly of / , ± / 5 2 1 2 symmetry. This is in good agreement with the anisotropy of the low-temperature magnetization 28 and susceptibility 53 , thereby supporting also the results of LDA + DMFT calculations at 290 K 54 .
We now will use density functional theory based band structure calculations to obtain a deeper understanding of the experimental findings (details in Section Materials and Methods, Band structure calculations). In Fig. 5(a) we display our result of a full-relativistic non-magnetic calculation employing the local density approximation (LDA) for CeRu 4 Sn 6 . We can clearly observe that the band structure has a true band gap in agreement with an earlier study in ref. 29, comprising of unoccupied Ce 4f bands above the Fermi level and occupied Ru 4d bands below the Fermi level. The presence of the gap is important since it confirms that the system has the even number of electrons per unit cell necessary to allow for the formation of a fully spin-compensated, non-magnetic state. We also notice in the left panel of Fig. 5(a) that the lowest unoccupied Ce 4f state is surprisingly given by the = / , ± / J 5 2 3 2 z (blue dots) and not by the / , ± / 5 2 1 2 (red dots). Yet, looking at the Ce 4f partial density of states depicted in the right panel of Fig. 5(a) we find that it is the / , ± / 5 2 1 2 which has the largest occupation, as found in our experiment. We therefore can infer from these calculations that the empty / , ± / 5 2 1 2 has indeed the stronger hybridization with the occupied Ru 4d bands so that it gets pushed up higher in energy and, at the same time, acquires a larger occupation.
Next, we will make an attempt to estimate the effect of correlations on the relevant symmetries of the system. We do this by introducing artificially an extra potential for a particular J z state so that it becomes more occupied, i.e. as to mimic the Ce 4f 1 valence state thereby making the inversion with the Ru 4d. Figure 5 . We can observe that in the = / , ± / J 5 2 3 2 z case the system becomes metallic with quite a large number of bands crossing the Fermi level 55 . For the the / , ± / 5 2 1 2 case, we also see that the system is metallic, but only with a few crossings and small Fermi surface areas, suggesting that the number of charge carriers will be low. Important for the topology is that the band structure is that of a semi-metal, i.e. that there is a warped gap, so that a unique assignment can be made about the character of the bands. This aspect did not necessarily follow from the conditions set by Dzero and Galitzki 7,8 , since there could always have been some other non-bonding bands crossing the Fermi level, which is not considered in their effective model. Yet, our calculations indicate specifically that such is not the case for CeRu 4 Sn 6 . It is also important to notice that the warped gap is substantial, i.e. 20 meV or larger, which corroborates nicely with the fact that the J z = ± 1/2 is nodeless and therefore guarantees the presence of hybridization with the Ru 4d bands at all k-points of the Brillouin Zone (except at the time reversal inversion points).
To summarize, we thus have found that CeRu 4 Sn 6 fulfills the conditions mentioned by Dzero and Galitzki 7,8 to be a strong topological insulator. It has a ground state symmetry given primarily by the Γ = / , ± / 5 2 1 2 6 and has a 4f occupation sufficiently far away from integer with a very high Kondo temperature (170 K) so that the effective degeneracy is larger than two. We have in addition verified that there are no non-bonding bands that otherwise could spoil the formation of a warped gap. We therefore conclude that CeRu 4 Sn 6 is a strongly correlated material with non-trivial topology.

Samples.
All experiments were performed on single crystals. Single crystals were grown with the self-flux floating-zone melting method using optical heating in a four-mirror furnace 56 . The quality of the crystals was checked with Laue and x-ray diffraction as well as scanning electron microscopy and energy dispersive x-ray spectroscopy, which show that the samples are single grain and single phase materials. The lattice parameters are in agreement with a = 6.8810 Å and c = 9.7520 Å as given in ref. 23. The single crystals were pre-aligned with a Laue camera. A peculiarity of the present tetragonal structure is that ≈ a c 2 within ≈ 0.2%. As a consequence the structure appears pseudo-cubic in Laue images and the final alignment of the single crystals had to be done via magnetization measurements 26 .
Set-ups of x-ray techniques. The PFY-XAS experiment was performed at the GALAXIES inelastic scattering endstation at the SOLEIL synchrotron in France 57 . The synchrotron radiation was monochromatized using a Si(111) nitrogen-cooled fixed-exit double-crystal monochromator (Δ E/E ≈ 1.4 × 10 −4 ), followed by a Pd-coated spherical collimating mirror. The x-rays were then focused to a spot size of 30 μm × 90 μm FWHM [vertical x horizontal] at the sample position by a 3:1 toroid Pd-coated mirror. The overall energy resolution for the experiment was measured to be 700 meV FWHM with about 1 × 10 13 photons/second on the sample. The incident x-ray energy was scanned across the Ce L 3 absorption edge (≈ 5723 eV) while monitoring the intensity of the Ce α L 1 de-excitation (≈ 4839 eV) using a Ge 331 spherical bent crystal analyzer (Bragg angle 80.8°, 1 m radius of curvature) in the Rowland circle geometry which focused the emitted x-rays onto a silicon avalanche photodiode. All measurements took place in a backscattering geometry and normalization was performed using a monitor placed just before the sample that recorded the scattered radiation from a Kapton foil placed in the beam path. The flight path was filled with helium gas to reduce the air scattering and the samples were cooled in a helium cryostat.
The XAS spectra were recorded at the synchrotron light source BESSY II using the UE46 PGM-1 undulator beamline. The total electron yield (TEY) was measured under ultra high vacuum of the order of 10 −10 mbar. The CeRu 4 Sn 6 single crystals were cleaved in situ to guarantee clean sample surfaces. The TEY signal was normalized to the incoming photon flux I 0 as measured at the refocusing mirror. The energy resolution at the cerium M 4,5 edges was set to 0.15 eV. The undulator combined with a normal incident measurement geometry allow a change in polarization without changing the probing spot on the sample surface. The crystals were oriented such that the two polarizations E [001] and ⊥ E [001], [001] being the long tetragonal axis, could be measured. Several crystals were measured or re-cleaved in order to assure the reproducibility of the data.
The NIXS measurements were performed at the beamline ID20 at ESRF. A single Si(111) monochromator set the incident energy to 9690 eV and the scattered intensity was analyzed by three columns of three Si(660) analyzers (in total nine) at in-plane scattering angles of 2Θ = 140°, 146°, and 153°, corresponding to an averaged momentum transfer of ≈ ( . ± . ) −  q 9 4 0 2 Å 1 . This configuration yields an instrumental energy resolution of FWHM = 1.3 eV and made the Ce N 4,5 edge measurable beyond the dipole limit 44,50,51 . Two crystals with polished (100)  [001] in Fig. 6. We obtain reduction factors of 41% for the 4f-4f and 20% for the 3d-4f Coulomb interaction while the spin orbit values remain unchanged within 1 or 2%. These are typical values for Ce intermetallic compounds [41][42][43] . For the reproduction of the polarized data, the CEF has to be taken into account. The calculations are purely ionic so that the spectral weights due to f 0 in the ground states and transitions to continuum states are not captured in the simulation.
The NIXS data were simulated using the full multiplet code Quanty 58 . A Gaussian and a Lorentzian broadening of FWHM = 1.3 eV and 0.2 eV, respectively, are assumed to account for the instrumental resolution and life time effects. The atomic Hartree-Fock values were adjusted via the peak positions, resulting in reductions of 30% and 20% for the 4f-4f and 4d-4f Coulomb interactions, respectively. The simulation procedure is explained in detail in ref. 44. Band structure calculations. Density functional theory based calculations were performed using the full-potential non-orthogonal local orbital code (FPLO) 59 employing the local density approximation (LDA) and including spin-orbit coupling. The band structure and density of states (DOS) are obtained from a carefully converged calculation with 1000 k-points in the irreducible wedge of the Brillouin zone. For easy comparison with literature, we have plotted the band structure along the k-path as described in ref. 54. To simulate the enhancement in the occupation of the various J z states, we have applied a constant potential shift to the on-site energies (1 eV) of the relevant orbitals.